| L(s) = 1 | − 2·5-s + 2·7-s − 6·11-s + 13-s + 2·17-s + 6·19-s − 25-s − 6·29-s + 6·31-s − 4·35-s − 2·37-s + 10·41-s − 8·43-s − 6·47-s − 3·49-s + 6·53-s + 12·55-s − 6·59-s + 10·61-s − 2·65-s − 2·67-s + 14·71-s − 14·73-s − 12·77-s + 4·79-s + 6·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 1.80·11-s + 0.277·13-s + 0.485·17-s + 1.37·19-s − 1/5·25-s − 1.11·29-s + 1.07·31-s − 0.676·35-s − 0.328·37-s + 1.56·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s + 0.824·53-s + 1.61·55-s − 0.781·59-s + 1.28·61-s − 0.248·65-s − 0.244·67-s + 1.66·71-s − 1.63·73-s − 1.36·77-s + 0.450·79-s + 0.658·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−7.72311637943970507489745672120, −7.15351246089745902827498765258, −6.01751931907727312393576018481, −5.25874893030557723454431240338, −4.87531570308063726295785778280, −3.88536518722750035916842998102, −3.17455449304663684213943314812, −2.34100159140218790887026848642, −1.18343602639981650227699912137, 0,
1.18343602639981650227699912137, 2.34100159140218790887026848642, 3.17455449304663684213943314812, 3.88536518722750035916842998102, 4.87531570308063726295785778280, 5.25874893030557723454431240338, 6.01751931907727312393576018481, 7.15351246089745902827498765258, 7.72311637943970507489745672120
