| L(s) = 1 | + 2-s − 4·3-s − 4·6-s + 8-s + 6·9-s − 6·11-s + 3·16-s + 6·18-s − 6·19-s − 6·22-s − 4·24-s − 9·25-s − 16·29-s + 6·31-s + 6·32-s + 24·33-s + 6·37-s − 6·38-s + 4·47-s − 12·48-s − 9·50-s − 4·53-s + 24·57-s − 16·58-s − 14·59-s − 12·61-s + 6·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s − 1.63·6-s + 0.353·8-s + 2·9-s − 1.80·11-s + 3/4·16-s + 1.41·18-s − 1.37·19-s − 1.27·22-s − 0.816·24-s − 9/5·25-s − 2.97·29-s + 1.07·31-s + 1.06·32-s + 4.17·33-s + 0.986·37-s − 0.973·38-s + 0.583·47-s − 1.73·48-s − 1.27·50-s − 0.549·53-s + 3.17·57-s − 2.10·58-s − 1.82·59-s − 1.53·61-s + 0.762·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2124686658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2124686658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T + T^{2} - p T^{3} - p^{2} T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | \( ( 1 + T + T^{2} )^{4} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 9 T^{2} + 9 p T^{4} - 96 T^{5} + 66 T^{6} - 864 T^{7} - 394 T^{8} - 864 p T^{9} + 66 p^{2} T^{10} - 96 p^{3} T^{11} + 9 p^{5} T^{12} + 9 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 2 T + 25 T^{2} - 58 T^{3} + 372 T^{4} - 58 p T^{5} + 25 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )( 1 + 8 T + 34 T^{2} + 112 T^{3} + 339 T^{4} + 112 p T^{5} + 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 13 | \( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 49362 p^{2} T^{10} + 2241 p^{4} T^{12} - 66 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 40 T^{2} + 786 T^{4} - 1104 T^{5} + 10208 T^{6} - 39072 T^{7} + 129731 T^{8} - 39072 p T^{9} + 10208 p^{2} T^{10} - 1104 p^{3} T^{11} + 786 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 + 6 T - 33 T^{2} - 150 T^{3} + 1165 T^{4} + 1968 T^{5} - 34182 T^{6} - 11868 T^{7} + 759066 T^{8} - 11868 p T^{9} - 34182 p^{2} T^{10} + 1968 p^{3} T^{11} + 1165 p^{4} T^{12} - 150 p^{5} T^{13} - 33 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 + 52 T^{2} + 54 p T^{4} - 2784 T^{5} + 24080 T^{6} - 140352 T^{7} + 497843 T^{8} - 140352 p T^{9} + 24080 p^{2} T^{10} - 2784 p^{3} T^{11} + 54 p^{5} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 8 T + 71 T^{2} + 344 T^{3} + 1924 T^{4} + 344 p T^{5} + 71 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 6 T - 4 T^{2} + 336 T^{3} - 2729 T^{4} + 10764 T^{5} + 2216 T^{6} - 444234 T^{7} + 3877768 T^{8} - 444234 p T^{9} + 2216 p^{2} T^{10} + 10764 p^{3} T^{11} - 2729 p^{4} T^{12} + 336 p^{5} T^{13} - 4 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 6 T - 69 T^{2} - 18 T^{3} + 4753 T^{4} + 9780 T^{5} - 152586 T^{6} - 146184 T^{7} + 2893194 T^{8} - 146184 p T^{9} - 152586 p^{2} T^{10} + 9780 p^{3} T^{11} + 4753 p^{4} T^{12} - 18 p^{5} T^{13} - 69 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 607176 p^{2} T^{10} + 9948 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 1729746 p^{2} T^{10} + 23793 p^{4} T^{12} - 210 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 4 T - 144 T^{2} + 456 T^{3} + 12722 T^{4} - 27948 T^{5} - 805600 T^{6} + 556940 T^{7} + 41968563 T^{8} + 556940 p T^{9} - 805600 p^{2} T^{10} - 27948 p^{3} T^{11} + 12722 p^{4} T^{12} + 456 p^{5} T^{13} - 144 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 + 4 T - 135 T^{2} - 900 T^{3} + 9413 T^{4} + 70368 T^{5} - 312982 T^{6} - 1938608 T^{7} + 10598262 T^{8} - 1938608 p T^{9} - 312982 p^{2} T^{10} + 70368 p^{3} T^{11} + 9413 p^{4} T^{12} - 900 p^{5} T^{13} - 135 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 14 T - 13 T^{2} - 1110 T^{3} - 3463 T^{4} + 11848 T^{5} - 87914 T^{6} + 1416852 T^{7} + 32978194 T^{8} + 1416852 p T^{9} - 87914 p^{2} T^{10} + 11848 p^{3} T^{11} - 3463 p^{4} T^{12} - 1110 p^{5} T^{13} - 13 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 12 T + 180 T^{2} + 1584 T^{3} + 12426 T^{4} + 46860 T^{5} + 10032 T^{6} - 3576756 T^{7} - 33274477 T^{8} - 3576756 p T^{9} + 10032 p^{2} T^{10} + 46860 p^{3} T^{11} + 12426 p^{4} T^{12} + 1584 p^{5} T^{13} + 180 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 42 T + 1023 T^{2} + 18270 T^{3} + 261141 T^{4} + 3133152 T^{5} + 32837970 T^{6} + 307895844 T^{7} + 2631022010 T^{8} + 307895844 p T^{9} + 32837970 p^{2} T^{10} + 3133152 p^{3} T^{11} + 261141 p^{4} T^{12} + 18270 p^{5} T^{13} + 1023 p^{6} T^{14} + 42 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 4336608 p^{2} T^{10} + 41724 p^{4} T^{12} - 288 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 18 T + 347 T^{2} - 4302 T^{3} + 50601 T^{4} - 463140 T^{5} + 4384558 T^{6} - 34967736 T^{7} + 313616978 T^{8} - 34967736 p T^{9} + 4384558 p^{2} T^{10} - 463140 p^{3} T^{11} + 50601 p^{4} T^{12} - 4302 p^{5} T^{13} + 347 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 - 6 T + 132 T^{2} - 720 T^{3} + 9999 T^{4} - 52644 T^{5} - 314880 T^{6} - 1496442 T^{7} - 48671848 T^{8} - 1496442 p T^{9} - 314880 p^{2} T^{10} - 52644 p^{3} T^{11} + 9999 p^{4} T^{12} - 720 p^{5} T^{13} + 132 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 2 T + 229 T^{2} - 802 T^{3} + 24432 T^{4} - 802 p T^{5} + 229 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 264 T^{2} + 38610 T^{4} - 50400 T^{5} + 4052064 T^{6} - 8952768 T^{7} + 353995811 T^{8} - 8952768 p T^{9} + 4052064 p^{2} T^{10} - 50400 p^{3} T^{11} + 38610 p^{4} T^{12} + 264 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 28537554 p^{2} T^{10} + 165777 p^{4} T^{12} - 594 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.63884515983790031468666309169, −4.52526432535786638091686039826, −4.46765348884984083116591223981, −4.38571792560063248575416743454, −4.37944607436527994133678185294, −4.02258909520958459985856958264, −3.96788434841005741040500575472, −3.71042282141582152717911349319, −3.60859987240446128047443573699, −3.48203655935270285514973492007, −3.14442011355866459505568198205, −3.05341118082382707440408577660, −2.96366112504726810216236456762, −2.87463584736089871796720205529, −2.51352636906772470016397190993, −2.51242558148729858285413205028, −2.16702192549783103258512792156, −1.91983390417861518422638830240, −1.91022711845055511818851148000, −1.70259850296755762347576629265, −1.35565825997104066407412140908, −1.26172692479134175700682055139, −0.77035298647698130760441414687, −0.37269891647278455313406012649, −0.16250443039731284826946977802,
0.16250443039731284826946977802, 0.37269891647278455313406012649, 0.77035298647698130760441414687, 1.26172692479134175700682055139, 1.35565825997104066407412140908, 1.70259850296755762347576629265, 1.91022711845055511818851148000, 1.91983390417861518422638830240, 2.16702192549783103258512792156, 2.51242558148729858285413205028, 2.51352636906772470016397190993, 2.87463584736089871796720205529, 2.96366112504726810216236456762, 3.05341118082382707440408577660, 3.14442011355866459505568198205, 3.48203655935270285514973492007, 3.60859987240446128047443573699, 3.71042282141582152717911349319, 3.96788434841005741040500575472, 4.02258909520958459985856958264, 4.37944607436527994133678185294, 4.38571792560063248575416743454, 4.46765348884984083116591223981, 4.52526432535786638091686039826, 4.63884515983790031468666309169
Plot not available for L-functions of degree greater than 10.
