| L(s) = 1 | − 2·5-s + 7-s − 3·9-s + 2·13-s + 17-s − 4·19-s − 25-s + 6·29-s − 2·35-s + 6·37-s − 6·41-s + 12·43-s + 6·45-s + 8·47-s + 49-s + 2·53-s − 4·59-s − 2·61-s − 3·63-s − 4·65-s − 12·67-s + 2·73-s − 8·79-s + 9·81-s − 12·83-s − 2·85-s + 10·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 9-s + 0.554·13-s + 0.242·17-s − 0.917·19-s − 1/5·25-s + 1.11·29-s − 0.338·35-s + 0.986·37-s − 0.937·41-s + 1.82·43-s + 0.894·45-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.520·59-s − 0.256·61-s − 0.377·63-s − 0.496·65-s − 1.46·67-s + 0.234·73-s − 0.900·79-s + 81-s − 1.31·83-s − 0.216·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−7.70887399734067596807145353717, −6.87833629851226703985750969896, −6.05001424602115039966589390130, −5.55335669661380124670059164485, −4.48098046695808625370277161597, −4.06274349339875145917341865187, −3.10438371895445250824869484753, −2.39660886070895991929260532625, −1.13629511962707899191732721753, 0,
1.13629511962707899191732721753, 2.39660886070895991929260532625, 3.10438371895445250824869484753, 4.06274349339875145917341865187, 4.48098046695808625370277161597, 5.55335669661380124670059164485, 6.05001424602115039966589390130, 6.87833629851226703985750969896, 7.70887399734067596807145353717
