| L(s) = 1 | − 4·2-s + 4·3-s + 10·4-s + 5-s − 16·6-s − 4·7-s − 20·8-s + 10·9-s − 4·10-s − 4·11-s + 40·12-s − 3·13-s + 16·14-s + 4·15-s + 35·16-s − 3·17-s − 40·18-s − 4·19-s + 10·20-s − 16·21-s + 16·22-s + 10·23-s − 80·24-s − 13·25-s + 12·26-s + 20·27-s − 40·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s + 0.447·5-s − 6.53·6-s − 1.51·7-s − 7.07·8-s + 10/3·9-s − 1.26·10-s − 1.20·11-s + 11.5·12-s − 0.832·13-s + 4.27·14-s + 1.03·15-s + 35/4·16-s − 0.727·17-s − 9.42·18-s − 0.917·19-s + 2.23·20-s − 3.49·21-s + 3.41·22-s + 2.08·23-s − 16.3·24-s − 2.59·25-s + 2.35·26-s + 3.84·27-s − 7.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 61 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - T + 14 T^{2} - 14 T^{3} + 96 T^{4} - 14 p T^{5} + 14 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 4 T + 27 T^{2} + 69 T^{3} + 272 T^{4} + 69 p T^{5} + 27 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 3 T + 20 T^{2} - 38 T^{3} + 4 T^{4} - 38 p T^{5} + 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 3 T + 37 T^{2} + 122 T^{3} + 674 T^{4} + 122 p T^{5} + 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 4 T + 69 T^{2} + 213 T^{3} + 1898 T^{4} + 213 p T^{5} + 69 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 10 T + 89 T^{2} - 413 T^{3} + 2352 T^{4} - 413 p T^{5} + 89 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 10 T + 127 T^{2} + 793 T^{3} + 5546 T^{4} + 793 p T^{5} + 127 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 9 T + 88 T^{2} + 326 T^{3} + 2380 T^{4} + 326 p T^{5} + 88 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 6 T + 69 T^{2} + 397 T^{3} + 3722 T^{4} + 397 p T^{5} + 69 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 3 T + 115 T^{2} - 286 T^{3} + 6044 T^{4} - 286 p T^{5} + 115 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 18 T + 225 T^{2} + 2065 T^{3} + 15776 T^{4} + 2065 p T^{5} + 225 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 21 T + 304 T^{2} - 3062 T^{3} + 24182 T^{4} - 3062 p T^{5} + 304 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 20 T + 295 T^{2} + 3005 T^{3} + 25640 T^{4} + 3005 p T^{5} + 295 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 5 T + 18 T^{2} + 2 T^{3} + 3718 T^{4} + 2 p T^{5} + 18 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 3 T + 266 T^{2} + 598 T^{3} + 26668 T^{4} + 598 p T^{5} + 266 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 9 T + 244 T^{2} + 1588 T^{3} + 24338 T^{4} + 1588 p T^{5} + 244 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 51 T^{2} - 377 T^{3} + 7094 T^{4} - 377 p T^{5} + 51 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 31 T + 594 T^{2} + 7924 T^{3} + 81606 T^{4} + 7924 p T^{5} + 594 p^{2} T^{6} + 31 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 8 T + 267 T^{2} + 1711 T^{3} + 30448 T^{4} + 1711 p T^{5} + 267 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 5 T + 242 T^{2} - 1162 T^{3} + 28506 T^{4} - 1162 p T^{5} + 242 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 25 T + 412 T^{2} + 5726 T^{3} + 65012 T^{4} + 5726 p T^{5} + 412 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.65095096406869249314131224096, −6.25118974131979911294601790326, −6.03881077407887732384754948558, −5.92902107815683103069277943554, −5.87198498237554028798472281164, −5.24906402405690229715913122544, −5.18929769235158963765205424765, −5.09265764639586087322230937848, −5.07511229753453376453874336349, −4.18825423089404932833624306828, −4.18499852236459997149908716474, −4.12574804136113949822176384466, −3.80842564834959520680796714861, −3.50514389383045682347497643772, −3.23746890373320476897455512096, −3.16611622088774209484239955441, −2.93063895946999097653303213421, −2.54247979160432359501931285506, −2.50629131370288968119350801176, −2.47802105401937259561813189382, −2.06348580908651858049709033091, −1.75947534261781835650324793464, −1.54950363783201498041551492177, −1.34375846112533918275294567095, −1.30257896553921120174297741290, 0, 0, 0, 0,
1.30257896553921120174297741290, 1.34375846112533918275294567095, 1.54950363783201498041551492177, 1.75947534261781835650324793464, 2.06348580908651858049709033091, 2.47802105401937259561813189382, 2.50629131370288968119350801176, 2.54247979160432359501931285506, 2.93063895946999097653303213421, 3.16611622088774209484239955441, 3.23746890373320476897455512096, 3.50514389383045682347497643772, 3.80842564834959520680796714861, 4.12574804136113949822176384466, 4.18499852236459997149908716474, 4.18825423089404932833624306828, 5.07511229753453376453874336349, 5.09265764639586087322230937848, 5.18929769235158963765205424765, 5.24906402405690229715913122544, 5.87198498237554028798472281164, 5.92902107815683103069277943554, 6.03881077407887732384754948558, 6.25118974131979911294601790326, 6.65095096406869249314131224096
