| L(s) = 1 | − 2·2-s − 4-s + 16·7-s − 4·8-s − 22·11-s − 80·13-s − 32·14-s − 19·16-s − 164·17-s − 36·19-s + 44·22-s − 172·23-s + 160·26-s − 16·28-s + 108·29-s − 448·31-s + 202·32-s + 328·34-s − 108·37-s + 72·38-s + 212·41-s − 156·43-s + 22·44-s + 344·46-s + 20·47-s − 194·49-s + 80·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/8·4-s + 0.863·7-s − 0.176·8-s − 0.603·11-s − 1.70·13-s − 0.610·14-s − 0.296·16-s − 2.33·17-s − 0.434·19-s + 0.426·22-s − 1.55·23-s + 1.20·26-s − 0.107·28-s + 0.691·29-s − 2.59·31-s + 1.11·32-s + 1.65·34-s − 0.479·37-s + 0.307·38-s + 0.807·41-s − 0.553·43-s + 0.0753·44-s + 1.10·46-s + 0.0620·47-s − 0.565·49-s + 0.213·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1222837311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1222837311\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 5 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 450 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 4542 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 164 T + 16118 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 36 T + 3242 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 172 T + 30758 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 108 T + 14062 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 448 T + 101646 T^{2} + 448 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 108 T + 103022 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 212 T + 148886 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 156 T + 63530 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 20 T + 202454 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 132 T - 117518 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 688 T + 404246 T^{2} - 688 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 96 T + 402398 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 448 T + 455094 T^{2} + 448 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 132 T - 187322 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 428 T + 808278 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 424 T + 1031010 T^{2} - 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 720 T + 1262374 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1056 T + 1317010 T^{2} - 1056 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 52 T - 413466 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.622110113012283536957003254808, −8.586709407531431339843037955841, −8.054270367349900859201845546046, −7.74169854110407557499197070317, −7.18869066002806614818232801798, −7.07979019644320818527655992847, −6.59959456943534649435863944904, −6.00793498418536416331862510805, −5.77344501637341867988289807565, −5.07449417651987644173873234832, −4.78627915260909653607843946762, −4.56148365162648233350493963957, −4.04355402912035458739786677112, −3.57618862223068357599214026297, −2.78660034132264826234656391960, −2.28244223881621095808931137545, −2.08119317437813433496229211913, −1.70356654271852401246880771492, −0.59963214944309582194366329681, −0.11756153249631857978140265156,
0.11756153249631857978140265156, 0.59963214944309582194366329681, 1.70356654271852401246880771492, 2.08119317437813433496229211913, 2.28244223881621095808931137545, 2.78660034132264826234656391960, 3.57618862223068357599214026297, 4.04355402912035458739786677112, 4.56148365162648233350493963957, 4.78627915260909653607843946762, 5.07449417651987644173873234832, 5.77344501637341867988289807565, 6.00793498418536416331862510805, 6.59959456943534649435863944904, 7.07979019644320818527655992847, 7.18869066002806614818232801798, 7.74169854110407557499197070317, 8.054270367349900859201845546046, 8.586709407531431339843037955841, 8.622110113012283536957003254808
