| L(s) = 1 | − 2-s − 3·5-s + 3·10-s + 2·11-s − 4·13-s − 16-s − 2·17-s − 7·19-s − 2·22-s − 3·23-s − 25-s + 4·26-s − 29-s − 3·31-s + 32-s + 2·34-s + 10·37-s + 7·38-s − 16·41-s + 3·43-s + 3·46-s + 5·47-s + 50-s − 5·53-s − 6·55-s + 58-s − 20·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s + 0.948·10-s + 0.603·11-s − 1.10·13-s − 1/4·16-s − 0.485·17-s − 1.60·19-s − 0.426·22-s − 0.625·23-s − 1/5·25-s + 0.784·26-s − 0.185·29-s − 0.538·31-s + 0.176·32-s + 0.342·34-s + 1.64·37-s + 1.13·38-s − 2.49·41-s + 0.457·43-s + 0.442·46-s + 0.729·47-s + 0.141·50-s − 0.686·53-s − 0.809·55-s + 0.131·58-s − 2.60·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + T + T^{2} + T^{3} + p T^{4} + p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 3 T + 2 p T^{2} + p^{2} T^{3} + 74 T^{4} + p^{3} T^{5} + 2 p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 2 T + 20 T^{2} - 34 T^{3} + 294 T^{4} - 34 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 40 T^{2} + 62 T^{3} + 878 T^{4} + 62 p T^{5} + 40 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 7 T + 64 T^{2} + 351 T^{3} + 1774 T^{4} + 351 p T^{5} + 64 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 3 T + 40 T^{2} - 49 T^{3} + 494 T^{4} - 49 p T^{5} + 40 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + T + 86 T^{2} + 35 T^{3} + 3378 T^{4} + 35 p T^{5} + 86 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 3 T - 4 T^{2} + 119 T^{3} + 58 p T^{4} + 119 p T^{5} - 4 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 10 T + 64 T^{2} - 270 T^{3} + 1870 T^{4} - 270 p T^{5} + 64 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 16 T + 4 p T^{2} + 1280 T^{3} + 8694 T^{4} + 1280 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 3 T + 128 T^{2} - 275 T^{3} + 7246 T^{4} - 275 p T^{5} + 128 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 5 T + 148 T^{2} - 689 T^{3} + 9638 T^{4} - 689 p T^{5} + 148 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 5 T + 174 T^{2} + 727 T^{3} + 12802 T^{4} + 727 p T^{5} + 174 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 20 T + 316 T^{2} + 3236 T^{3} + 28790 T^{4} + 3236 p T^{5} + 316 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 12 T + 180 T^{2} + 1508 T^{3} + 15014 T^{4} + 1508 p T^{5} + 180 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 22 T + 228 T^{2} + 1254 T^{3} + 6086 T^{4} + 1254 p T^{5} + 228 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 52 T^{2} - 304 T^{3} + 7478 T^{4} - 304 p T^{5} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 13 T + 126 T^{2} + 261 T^{3} - 3934 T^{4} + 261 p T^{5} + 126 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 11 T + 196 T^{2} - 1167 T^{3} + 15030 T^{4} - 1167 p T^{5} + 196 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - T + 296 T^{2} - 329 T^{3} + 35310 T^{4} - 329 p T^{5} + 296 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 5 T + 194 T^{2} + 139 T^{3} + 16986 T^{4} + 139 p T^{5} + 194 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 17 T + 374 T^{2} - 4127 T^{3} + 52210 T^{4} - 4127 p T^{5} + 374 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.22990041440798402899152382151, −5.86911447022884152008077434673, −5.69209077603490395745469986542, −5.53795868160919085390447662320, −5.45917633279771461146456915261, −4.99353186951057381344112732701, −4.93839525140265462414638901037, −4.64441805420676099603666516172, −4.56729295247629563909694239013, −4.20542055827670338463232149552, −4.15250843735192242550808251397, −4.04982139901124565441565025778, −4.02809745942750714628333507919, −3.47090250326482962683171520744, −3.40650829547879088130661958246, −3.01951775836285878222624563232, −2.93962049313617378947003947423, −2.85489606518411439928105635833, −2.28927227031623131388669179107, −2.21237405843254983204492030968, −2.02128732798285343792710446009, −1.68795370205762329259032125000, −1.50751898291570420595350312586, −1.11834739631990087472559495899, −0.917038887603706134983942892517, 0, 0, 0, 0,
0.917038887603706134983942892517, 1.11834739631990087472559495899, 1.50751898291570420595350312586, 1.68795370205762329259032125000, 2.02128732798285343792710446009, 2.21237405843254983204492030968, 2.28927227031623131388669179107, 2.85489606518411439928105635833, 2.93962049313617378947003947423, 3.01951775836285878222624563232, 3.40650829547879088130661958246, 3.47090250326482962683171520744, 4.02809745942750714628333507919, 4.04982139901124565441565025778, 4.15250843735192242550808251397, 4.20542055827670338463232149552, 4.56729295247629563909694239013, 4.64441805420676099603666516172, 4.93839525140265462414638901037, 4.99353186951057381344112732701, 5.45917633279771461146456915261, 5.53795868160919085390447662320, 5.69209077603490395745469986542, 5.86911447022884152008077434673, 6.22990041440798402899152382151
