| L(s) = 1 | − 4·2-s − 18·3-s + 4·4-s − 48·5-s + 72·6-s + 72·7-s − 96·8-s + 243·9-s + 192·10-s − 596·11-s − 72·12-s − 338·13-s − 288·14-s + 864·15-s − 176·16-s − 268·17-s − 972·18-s + 1.12e3·19-s − 192·20-s − 1.29e3·21-s + 2.38e3·22-s − 1.76e3·23-s + 1.72e3·24-s − 3.12e3·25-s + 1.35e3·26-s − 2.91e3·27-s + 288·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/8·4-s − 0.858·5-s + 0.816·6-s + 0.555·7-s − 0.530·8-s + 9-s + 0.607·10-s − 1.48·11-s − 0.144·12-s − 0.554·13-s − 0.392·14-s + 0.991·15-s − 0.171·16-s − 0.224·17-s − 0.707·18-s + 0.716·19-s − 0.107·20-s − 0.641·21-s + 1.05·22-s − 0.696·23-s + 0.612·24-s − 0.999·25-s + 0.392·26-s − 0.769·27-s + 0.0694·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p^{2} T + 3 p^{2} T^{2} + p^{7} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 48 T + 5426 T^{2} + 48 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 72 T + 28134 T^{2} - 72 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 596 T + 312122 T^{2} + 596 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 268 T + 2228454 T^{2} + 268 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1128 T + 4971870 T^{2} - 1128 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1768 T + 12073598 T^{2} + 1768 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7612 T + 55112798 T^{2} + 7612 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4160 T + 24205318 T^{2} + 4160 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 17468 T + 184139086 T^{2} - 17468 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 28000 T + 412511706 T^{2} + 28000 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 24328 T + 8556274 p T^{2} + 24328 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18108 T + 278422754 T^{2} + 18108 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1420 T + 800695790 T^{2} - 1420 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6788 T - 153114342 T^{2} - 6788 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 37148 T + 1888667614 T^{2} + 37148 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 106112 T + 5372723950 T^{2} - 106112 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 30460 T + 2533993202 T^{2} + 30460 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 37620 T + 1766924310 T^{2} + 37620 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2160 T + 3629418462 T^{2} + 2160 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 207004 T + 18578057034 T^{2} - 207004 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 74136 T + 5425645898 T^{2} + 74136 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 121156 T + 20409714022 T^{2} + 121156 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.16120570322920066907942981407, −14.65906338311515355859574003103, −13.29635107299606441806024239338, −13.28382821704473411585589058138, −12.23227229682880046739953021514, −11.55841653714141100940525583543, −11.48534131781404559461556303851, −10.74056207993962117380546028237, −9.746223423261678236679090479795, −9.668030029684409238596506662149, −8.223315267029719505497435667540, −7.969677681716994230284894042754, −7.21971667344088841155056852294, −6.29205901625479135001414939731, −5.31103400153986507010528407785, −4.78918212884744812920962864226, −3.51787287145009873692907807117, −1.93562694482452864905691269086, 0, 0,
1.93562694482452864905691269086, 3.51787287145009873692907807117, 4.78918212884744812920962864226, 5.31103400153986507010528407785, 6.29205901625479135001414939731, 7.21971667344088841155056852294, 7.969677681716994230284894042754, 8.223315267029719505497435667540, 9.668030029684409238596506662149, 9.746223423261678236679090479795, 10.74056207993962117380546028237, 11.48534131781404559461556303851, 11.55841653714141100940525583543, 12.23227229682880046739953021514, 13.28382821704473411585589058138, 13.29635107299606441806024239338, 14.65906338311515355859574003103, 15.16120570322920066907942981407
