| L(s) = 1 | − 5-s + 3·7-s − 3·11-s − 13-s + 17-s + 6·19-s − 5·23-s + 25-s + 6·29-s − 2·31-s − 3·35-s + 7·37-s − 3·41-s + 8·43-s − 2·47-s + 2·49-s + 53-s + 3·55-s − 15·61-s + 65-s − 12·67-s + 5·71-s − 6·73-s − 9·77-s + 13·79-s + 12·83-s − 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s − 0.904·11-s − 0.277·13-s + 0.242·17-s + 1.37·19-s − 1.04·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s − 0.507·35-s + 1.15·37-s − 0.468·41-s + 1.21·43-s − 0.291·47-s + 2/7·49-s + 0.137·53-s + 0.404·55-s − 1.92·61-s + 0.124·65-s − 1.46·67-s + 0.593·71-s − 0.702·73-s − 1.02·77-s + 1.46·79-s + 1.31·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.003900205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.003900205\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−7.80860383591300069581567959531, −7.32957585345103008421660254266, −6.26567235789402417215419342910, −5.56440328248277854757199292242, −4.85862628025655817297060772428, −4.41029364293374028330845727703, −3.38521489853123041301970159108, −2.64340422040986262164550019090, −1.71022288292620648798471288509, −0.69174389754390580558784752790,
0.69174389754390580558784752790, 1.71022288292620648798471288509, 2.64340422040986262164550019090, 3.38521489853123041301970159108, 4.41029364293374028330845727703, 4.85862628025655817297060772428, 5.56440328248277854757199292242, 6.26567235789402417215419342910, 7.32957585345103008421660254266, 7.80860383591300069581567959531
