| L(s) = 1 | + 5-s − 3·7-s + 3·11-s + 3·17-s − 5·19-s − 3·23-s + 25-s + 5·29-s − 3·35-s − 11·37-s + 5·41-s − 7·43-s + 8·47-s + 2·49-s + 10·53-s + 3·55-s + 59-s − 5·61-s + 11·67-s − 3·71-s − 6·73-s − 9·77-s + 16·83-s + 3·85-s + 13·89-s − 5·95-s − 7·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 0.904·11-s + 0.727·17-s − 1.14·19-s − 0.625·23-s + 1/5·25-s + 0.928·29-s − 0.507·35-s − 1.80·37-s + 0.780·41-s − 1.06·43-s + 1.16·47-s + 2/7·49-s + 1.37·53-s + 0.404·55-s + 0.130·59-s − 0.640·61-s + 1.34·67-s − 0.356·71-s − 0.702·73-s − 1.02·77-s + 1.75·83-s + 0.325·85-s + 1.37·89-s − 0.512·95-s − 0.710·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−14.56274936117247, −13.85336844735148, −13.71740214338548, −12.99110936178738, −12.51408495763901, −12.05964339277820, −11.75013421468679, −10.79178087022086, −10.35132213720622, −10.08283746251689, −9.314522685009025, −9.066404660360468, −8.424846479604267, −7.866152887852278, −7.027522759381802, −6.598278042384818, −6.298015266977554, −5.598669694232354, −5.088572145628193, −4.135565779152164, −3.818817644064937, −3.110080598340847, −2.449853188827340, −1.738208269664773, −0.9272916187135996, 0,
0.9272916187135996, 1.738208269664773, 2.449853188827340, 3.110080598340847, 3.818817644064937, 4.135565779152164, 5.088572145628193, 5.598669694232354, 6.298015266977554, 6.598278042384818, 7.027522759381802, 7.866152887852278, 8.424846479604267, 9.066404660360468, 9.314522685009025, 10.08283746251689, 10.35132213720622, 10.79178087022086, 11.75013421468679, 12.05964339277820, 12.51408495763901, 12.99110936178738, 13.71740214338548, 13.85336844735148, 14.56274936117247
