| L(s) = 1 | − 2·3-s + 4·4-s − 3·9-s − 8·12-s − 64·13-s + 4·16-s + 4·19-s − 62·25-s + 62·27-s + 20·31-s − 12·36-s + 4·37-s + 128·39-s + 208·43-s − 8·48-s − 20·49-s − 256·52-s − 8·57-s + 212·61-s − 16·64-s + 156·67-s + 132·73-s + 124·75-s + 16·76-s − 52·79-s − 79·81-s − 40·93-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 4-s − 1/3·9-s − 2/3·12-s − 4.92·13-s + 1/4·16-s + 4/19·19-s − 2.47·25-s + 2.29·27-s + 0.645·31-s − 1/3·36-s + 4/37·37-s + 3.28·39-s + 4.83·43-s − 1/6·48-s − 0.408·49-s − 4.92·52-s − 0.140·57-s + 3.47·61-s − 1/4·64-s + 2.32·67-s + 1.80·73-s + 1.65·75-s + 4/19·76-s − 0.658·79-s − 0.975·81-s − 0.430·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8336775080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8336775080\) |
| \(L(2)\) |
|
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 | \( 1 + 2 T + 7 T^{2} - 14 p T^{3} - 4 p^{3} T^{4} - 14 p^{3} T^{5} + 7 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | \( ( 1 + 10 T^{2} + p^{4} T^{4} )^{2} \) |
| good | 5 | \( 1 + 62 T^{2} + 397 p T^{4} + 37758 T^{6} + 662756 T^{8} + 37758 p^{4} T^{10} + 397 p^{9} T^{12} + 62 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 - 70 T^{2} - 23407 T^{4} + 68250 T^{6} + 515186468 T^{8} + 68250 p^{4} T^{10} - 23407 p^{8} T^{12} - 70 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 16 T + 314 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 17 | \( 1 + 1118 T^{2} + 770753 T^{4} + 348960222 T^{6} + 118234187396 T^{8} + 348960222 p^{4} T^{10} + 770753 p^{8} T^{12} + 1118 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 2 T - 521 T^{2} + 394 T^{3} + 143860 T^{4} + 394 p^{2} T^{5} - 521 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 + 1370 T^{2} + 955793 T^{4} + 495152250 T^{6} + 244893547268 T^{8} + 495152250 p^{4} T^{10} + 955793 p^{8} T^{12} + 1370 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 764 T^{2} + 680486 T^{4} - 764 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 10 T - 1297 T^{2} + 5250 T^{3} + 931988 T^{4} + 5250 p^{2} T^{5} - 1297 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - T - 1368 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 5788 T^{2} + 13912710 T^{4} - 5788 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 52 T + 3582 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 47 | \( 1 + 538 T^{2} - 7938479 T^{4} - 823914182 T^{6} + 42475035844804 T^{8} - 823914182 p^{4} T^{10} - 7938479 p^{8} T^{12} + 538 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 + 5854 T^{2} + 10947457 T^{4} + 44144411038 T^{6} + 211248030023524 T^{8} + 44144411038 p^{4} T^{10} + 10947457 p^{8} T^{12} + 5854 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 + 746 T^{2} - 10773535 T^{4} - 9626884566 T^{6} - 25203887215996 T^{8} - 9626884566 p^{4} T^{10} - 10773535 p^{8} T^{12} + 746 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 106 T + 1337 T^{2} - 260442 T^{3} + 42335204 T^{4} - 260442 p^{2} T^{5} + 1337 p^{4} T^{6} - 106 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 78 T - 697 T^{2} + 171366 T^{3} - 1480236 T^{4} + 171366 p^{2} T^{5} - 697 p^{4} T^{6} - 78 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 1316 T^{2} + 44745734 T^{4} - 1316 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 66 T - 7039 T^{2} - 48642 T^{3} + 78234660 T^{4} - 48642 p^{2} T^{5} - 7039 p^{4} T^{6} - 66 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 26 T - 5617 T^{2} - 160914 T^{3} - 3567148 T^{4} - 160914 p^{2} T^{5} - 5617 p^{4} T^{6} + 26 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 18404 T^{2} + 168689894 T^{4} - 18404 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( 1 + 2542 T^{2} + 3560113 T^{4} - 311605556402 T^{6} - 4333594781349404 T^{8} - 311605556402 p^{4} T^{10} + 3560113 p^{8} T^{12} + 2542 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 - 144 T + 21802 T^{2} - 144 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−7.61336623382496843489784072779, −7.38700544920325456944915490509, −7.30434142130555478336228321573, −7.22693481106945414293389386481, −6.71414358444992867473273150842, −6.70092201068603518595443136175, −6.43221073517784398833118934603, −6.32911699900054416417586711327, −6.02115894214775578058465410881, −5.90008394344139750519219699834, −5.39911655415851667151436323962, −5.23014060394324740275620441639, −5.19748792837762772582726058973, −5.07098090224306135603626642584, −4.79927998329660573689296949336, −4.49722585801958726050108328778, −4.02915247494830154081848320648, −3.91588356256673611795640862293, −3.83322646708001216848073302714, −3.01201525965927102144182932257, −2.66180964612968656785158117705, −2.47624137765638921638372235460, −2.36855596773987885080361239393, −2.12843528403547952178587957365, −0.67547929323321649937980912680,
0.67547929323321649937980912680, 2.12843528403547952178587957365, 2.36855596773987885080361239393, 2.47624137765638921638372235460, 2.66180964612968656785158117705, 3.01201525965927102144182932257, 3.83322646708001216848073302714, 3.91588356256673611795640862293, 4.02915247494830154081848320648, 4.49722585801958726050108328778, 4.79927998329660573689296949336, 5.07098090224306135603626642584, 5.19748792837762772582726058973, 5.23014060394324740275620441639, 5.39911655415851667151436323962, 5.90008394344139750519219699834, 6.02115894214775578058465410881, 6.32911699900054416417586711327, 6.43221073517784398833118934603, 6.70092201068603518595443136175, 6.71414358444992867473273150842, 7.22693481106945414293389386481, 7.30434142130555478336228321573, 7.38700544920325456944915490509, 7.61336623382496843489784072779
Plot not available for L-functions of degree greater than 10.
