| L(s) = 1 | − 5-s − 3·7-s − 11-s − 13-s + 17-s + 2·19-s − 3·23-s + 25-s + 2·29-s + 6·31-s + 3·35-s + 11·37-s + 5·41-s − 4·43-s − 10·47-s + 2·49-s − 11·53-s + 55-s + 8·59-s + 13·61-s + 65-s − 12·67-s − 5·71-s + 10·73-s + 3·77-s + 3·79-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 0.301·11-s − 0.277·13-s + 0.242·17-s + 0.458·19-s − 0.625·23-s + 1/5·25-s + 0.371·29-s + 1.07·31-s + 0.507·35-s + 1.80·37-s + 0.780·41-s − 0.609·43-s − 1.45·47-s + 2/7·49-s − 1.51·53-s + 0.134·55-s + 1.04·59-s + 1.66·61-s + 0.124·65-s − 1.46·67-s − 0.593·71-s + 1.17·73-s + 0.341·77-s + 0.337·79-s − 1.31·83-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 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
−7.52145831062611022512156763785, −6.40778205427448072726373472276, −6.33859290798451640150635320062, −5.25582994582197556218619947582, −4.56986905156167203274964735994, −3.74101496794116888068356638855, −3.05934827518297177748040359480, −2.39612342515546914051064552815, −1.04685289311080159809055883415, 0,
1.04685289311080159809055883415, 2.39612342515546914051064552815, 3.05934827518297177748040359480, 3.74101496794116888068356638855, 4.56986905156167203274964735994, 5.25582994582197556218619947582, 6.33859290798451640150635320062, 6.40778205427448072726373472276, 7.52145831062611022512156763785
