| L(s) = 1 | − 2·7-s − 3·9-s − 4·11-s − 4·13-s − 6·17-s − 4·19-s − 2·23-s + 2·29-s − 4·31-s + 2·37-s − 2·41-s − 4·43-s + 2·47-s − 3·49-s + 6·53-s + 2·61-s + 6·63-s + 2·67-s + 8·71-s − 4·73-s + 8·77-s + 4·79-s + 9·81-s + 8·83-s − 2·89-s + 8·91-s + 6·97-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 9-s − 1.20·11-s − 1.10·13-s − 1.45·17-s − 0.917·19-s − 0.417·23-s + 0.371·29-s − 0.718·31-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 0.291·47-s − 3/7·49-s + 0.824·53-s + 0.256·61-s + 0.755·63-s + 0.244·67-s + 0.949·71-s − 0.468·73-s + 0.911·77-s + 0.450·79-s + 81-s + 0.878·83-s − 0.211·89-s + 0.838·91-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 229 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.08895591965599, −13.47688951848393, −13.08639656973153, −12.76808254887686, −12.10924428110392, −11.70530231546413, −11.05140551052225, −10.59033393244253, −10.27543728642449, −9.476548637674782, −9.255741834206200, −8.455723450597937, −8.206952797133837, −7.568295641839055, −6.845956057715518, −6.571109194241047, −5.877958685777910, −5.349237258124301, −4.836163301278906, −4.246846951557427, −3.513413979959148, −2.833092099941456, −2.375605577947474, −1.960413722979220, −0.4913078233249665, 0,
0.4913078233249665, 1.960413722979220, 2.375605577947474, 2.833092099941456, 3.513413979959148, 4.246846951557427, 4.836163301278906, 5.349237258124301, 5.877958685777910, 6.571109194241047, 6.845956057715518, 7.568295641839055, 8.206952797133837, 8.455723450597937, 9.255741834206200, 9.476548637674782, 10.27543728642449, 10.59033393244253, 11.05140551052225, 11.70530231546413, 12.10924428110392, 12.76808254887686, 13.08639656973153, 13.47688951848393, 14.08895591965599
