| L(s) = 1 | + 5-s − 7-s − 3·11-s + 4·13-s − 2·17-s + 4·19-s − 23-s − 4·25-s + 8·29-s − 4·31-s − 35-s − 7·37-s − 7·41-s − 2·43-s − 13·47-s − 6·49-s − 5·53-s − 3·55-s + 4·59-s + 4·61-s + 4·65-s − 12·67-s + 14·71-s − 11·73-s + 3·77-s + 4·79-s + 17·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.904·11-s + 1.10·13-s − 0.485·17-s + 0.917·19-s − 0.208·23-s − 4/5·25-s + 1.48·29-s − 0.718·31-s − 0.169·35-s − 1.15·37-s − 1.09·41-s − 0.304·43-s − 1.89·47-s − 6/7·49-s − 0.686·53-s − 0.404·55-s + 0.520·59-s + 0.512·61-s + 0.496·65-s − 1.46·67-s + 1.66·71-s − 1.28·73-s + 0.341·77-s + 0.450·79-s + 1.86·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 397 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.35430183274304, −12.80200991199244, −12.25606662423680, −11.68348758060404, −11.42258852632441, −10.73410857757158, −10.25724304885391, −10.11063543270134, −9.326895554736599, −9.106304105596348, −8.311916692958045, −8.123288816858536, −7.563892557016394, −6.817985662603782, −6.470350046182589, −6.087734299826195, −5.347514429518537, −5.079170496395671, −4.513657155658073, −3.617991014533697, −3.360766134019325, −2.825869974854277, −1.932159107126565, −1.678973430492477, −0.7480731863554876, 0,
0.7480731863554876, 1.678973430492477, 1.932159107126565, 2.825869974854277, 3.360766134019325, 3.617991014533697, 4.513657155658073, 5.079170496395671, 5.347514429518537, 6.087734299826195, 6.470350046182589, 6.817985662603782, 7.563892557016394, 8.123288816858536, 8.311916692958045, 9.106304105596348, 9.326895554736599, 10.11063543270134, 10.25724304885391, 10.73410857757158, 11.42258852632441, 11.68348758060404, 12.25606662423680, 12.80200991199244, 13.35430183274304
