| L(s) = 1 | − 5-s + 7-s − 4·11-s + 13-s − 6·17-s + 8·19-s − 4·23-s + 25-s − 2·29-s + 4·31-s − 35-s − 10·37-s + 2·41-s + 8·43-s + 8·47-s + 49-s + 6·53-s + 4·55-s − 8·59-s + 10·61-s − 65-s − 12·67-s − 8·71-s + 6·73-s − 4·77-s − 8·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.169·35-s − 1.64·37-s + 0.312·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s − 1.04·59-s + 1.28·61-s − 0.124·65-s − 1.46·67-s − 0.949·71-s + 0.702·73-s − 0.455·77-s − 0.900·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.17081216805704, −12.43534940404939, −12.15818685054833, −11.59243817682106, −11.27578555531452, −10.70134114012688, −10.36372214590793, −9.927152348871168, −9.147565509752840, −8.921191256210960, −8.340205133285325, −7.856601706530409, −7.362267396173986, −7.178573684619992, −6.384809780142566, −5.847465555813156, −5.294623242534797, −5.017169024211991, −4.188067827910523, −4.024193165303190, −3.150242126233133, −2.721805402820683, −2.152692580738213, −1.462954192881377, −0.6985594355912974, 0,
0.6985594355912974, 1.462954192881377, 2.152692580738213, 2.721805402820683, 3.150242126233133, 4.024193165303190, 4.188067827910523, 5.017169024211991, 5.294623242534797, 5.847465555813156, 6.384809780142566, 7.178573684619992, 7.362267396173986, 7.856601706530409, 8.340205133285325, 8.921191256210960, 9.147565509752840, 9.927152348871168, 10.36372214590793, 10.70134114012688, 11.27578555531452, 11.59243817682106, 12.15818685054833, 12.43534940404939, 13.17081216805704
