| L(s) = 1 | + 3-s + 4·5-s + 4·7-s + 9-s + 2·11-s + 4·15-s − 6·17-s − 4·19-s + 4·21-s + 4·23-s + 11·25-s + 27-s − 6·29-s − 8·31-s + 2·33-s + 16·35-s + 10·37-s + 4·41-s − 4·43-s + 4·45-s + 6·47-s + 9·49-s − 6·51-s + 6·53-s + 8·55-s − 4·57-s + 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 1.03·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 0.834·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.348·33-s + 2.70·35-s + 1.64·37-s + 0.624·41-s − 0.609·43-s + 0.596·45-s + 0.875·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s + 1.07·55-s − 0.529·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.169715580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.169715580\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.817407586522778069079144648315, −7.72879474900016567268618051227, −6.96083678200081693746573024083, −6.19828151787800730907119937682, −5.46198202571299738670788333206, −4.70278780739485109887488718416, −3.98300616324174991728330752059, −2.50958892989649883973601711340, −2.03328703404531547851260400557, −1.30627215120689870699242878095,
1.30627215120689870699242878095, 2.03328703404531547851260400557, 2.50958892989649883973601711340, 3.98300616324174991728330752059, 4.70278780739485109887488718416, 5.46198202571299738670788333206, 6.19828151787800730907119937682, 6.96083678200081693746573024083, 7.72879474900016567268618051227, 8.817407586522778069079144648315
