| L(s) = 1 | + 3-s − 2·5-s + 9-s + 4·11-s + 2·13-s − 2·15-s + 2·17-s − 8·23-s − 25-s + 27-s − 6·29-s − 8·31-s + 4·33-s − 6·37-s + 2·39-s + 6·41-s + 4·43-s − 2·45-s − 7·49-s + 2·51-s + 2·53-s − 8·55-s − 4·59-s − 2·61-s − 4·65-s + 4·67-s − 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s + 0.485·17-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 49-s + 0.280·51-s + 0.274·53-s − 1.07·55-s − 0.520·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 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 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−7.61950951666595077126724317917, −6.85891827919805546068039331531, −6.09791694801381232421198374318, −5.40997773154525098146238432632, −4.23309352016035067675133294948, −3.83764998170958613523904255647, −3.37632188765103348554746581671, −2.12120802782336302555457408398, −1.36402928363394079803758664889, 0,
1.36402928363394079803758664889, 2.12120802782336302555457408398, 3.37632188765103348554746581671, 3.83764998170958613523904255647, 4.23309352016035067675133294948, 5.40997773154525098146238432632, 6.09791694801381232421198374318, 6.85891827919805546068039331531, 7.61950951666595077126724317917
