| L(s) = 1 | + 4·4-s − 10·9-s + 80·13-s + 4·16-s + 64·19-s − 56·25-s + 128·31-s − 40·36-s − 80·37-s + 160·43-s + 320·52-s + 56·61-s − 16·64-s − 240·67-s − 120·73-s + 256·76-s + 128·79-s + 81·81-s − 720·97-s − 224·100-s + 160·103-s + 72·109-s − 800·117-s − 272·121-s + 512·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 4-s − 1.11·9-s + 6.15·13-s + 1/4·16-s + 3.36·19-s − 2.23·25-s + 4.12·31-s − 1.11·36-s − 2.16·37-s + 3.72·43-s + 6.15·52-s + 0.918·61-s − 1/4·64-s − 3.58·67-s − 1.64·73-s + 3.36·76-s + 1.62·79-s + 81-s − 7.42·97-s − 2.23·100-s + 1.55·103-s + 0.660·109-s − 6.83·117-s − 2.24·121-s + 4.12·124-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.09047270427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09047270427\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 + 10 T^{2} + 19 T^{4} + 10 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 56 T^{2} + 62 p^{2} T^{4} + 18816 T^{6} + 164771 T^{8} + 18816 p^{4} T^{10} + 62 p^{10} T^{12} + 56 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 + 272 T^{2} + 37406 T^{4} + 1984512 T^{6} + 40391459 T^{8} + 1984512 p^{4} T^{10} + 37406 p^{8} T^{12} + 272 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 20 T + 326 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 17 | \( 1 + 440 T^{2} - 626 p T^{4} + 16368000 T^{6} + 18569371523 T^{8} + 16368000 p^{4} T^{10} - 626 p^{9} T^{12} + 440 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 16 T - 105 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 - 688 T^{2} - 193474 T^{4} - 73709568 T^{6} + 233874201539 T^{8} - 73709568 p^{4} T^{10} - 193474 p^{8} T^{12} - 688 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 1652 T^{2} + 1917638 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 64 T + 1850 T^{2} - 20736 T^{3} + 184739 T^{4} - 20736 p^{2} T^{5} + 1850 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 20 T - 969 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 856 T^{2} - 2330094 T^{4} - 856 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 40 T + 3090 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 + 4346 T^{2} + 14008035 T^{4} + 4346 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 3026 T^{2} + 1266195 T^{4} + 3026 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 10532 T^{2} + 61824746 T^{4} + 261862971792 T^{6} + 919716139941299 T^{8} + 261862971792 p^{4} T^{10} + 61824746 p^{8} T^{12} + 10532 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 28 T - 4054 T^{2} + 72912 T^{3} + 6643139 T^{4} + 72912 p^{2} T^{5} - 4054 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 120 T + 1934 T^{2} + 418560 T^{3} + 70608435 T^{4} + 418560 p^{2} T^{5} + 1934 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 12176 T^{2} + 74167106 T^{4} - 12176 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 60 T - 6166 T^{2} - 53520 T^{3} + 47792115 T^{4} - 53520 p^{2} T^{5} - 6166 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 64 T - 6610 T^{2} + 113664 T^{3} + 50105219 T^{4} + 113664 p^{2} T^{5} - 6610 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 6356 T^{2} - 6983674 T^{4} - 6356 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( 1 + 27832 T^{2} + 456387886 T^{4} + 5364558328192 T^{6} + 48733823365562179 T^{8} + 5364558328192 p^{4} T^{10} + 456387886 p^{8} T^{12} + 27832 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 + 180 T + 26470 T^{2} + 180 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−5.18755010336256427673913783083, −4.76726144432924124374843298530, −4.72860292471311763533190879385, −4.59916203856295548703975743354, −4.34200518376308087807847974206, −4.02140322819766085413419924259, −3.99091994226387915465234989758, −3.92181221532003982712100595502, −3.82453030322951216894009503769, −3.60601008316721090468126863586, −3.40722436030286571332777434035, −3.12411329200142673020708730163, −3.12352061763795136720779507763, −3.02349357292368682893727408174, −2.85347970347065974018728409714, −2.47527038935002586427309043418, −2.34976938603797018483101980775, −2.21865308612847552681817271792, −1.83479038350015251795141285871, −1.29876680989642741932258011576, −1.21959708834471115213482437895, −1.14885814929620886645444178517, −1.12165303810224317291297010059, −1.07252272802622493402203191793, −0.02290256013664207553604224485,
0.02290256013664207553604224485, 1.07252272802622493402203191793, 1.12165303810224317291297010059, 1.14885814929620886645444178517, 1.21959708834471115213482437895, 1.29876680989642741932258011576, 1.83479038350015251795141285871, 2.21865308612847552681817271792, 2.34976938603797018483101980775, 2.47527038935002586427309043418, 2.85347970347065974018728409714, 3.02349357292368682893727408174, 3.12352061763795136720779507763, 3.12411329200142673020708730163, 3.40722436030286571332777434035, 3.60601008316721090468126863586, 3.82453030322951216894009503769, 3.92181221532003982712100595502, 3.99091994226387915465234989758, 4.02140322819766085413419924259, 4.34200518376308087807847974206, 4.59916203856295548703975743354, 4.72860292471311763533190879385, 4.76726144432924124374843298530, 5.18755010336256427673913783083
Plot not available for L-functions of degree greater than 10.
