| L(s) = 1 | + 2·5-s + 7-s − 3·9-s − 4·11-s − 13-s − 6·17-s − 8·23-s − 25-s − 10·29-s + 8·31-s + 2·35-s + 6·37-s − 6·41-s − 4·43-s − 6·45-s + 8·47-s + 49-s + 6·53-s − 8·55-s − 8·59-s + 10·61-s − 3·63-s − 2·65-s − 4·67-s + 8·71-s + 2·73-s − 4·77-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 9-s − 1.20·11-s − 0.277·13-s − 1.45·17-s − 1.66·23-s − 1/5·25-s − 1.85·29-s + 1.43·31-s + 0.338·35-s + 0.986·37-s − 0.937·41-s − 0.609·43-s − 0.894·45-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.07·55-s − 1.04·59-s + 1.28·61-s − 0.377·63-s − 0.248·65-s − 0.488·67-s + 0.949·71-s + 0.234·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 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 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−9.134931013069261030986986227797, −8.289068147988507628565322456813, −7.68484370188773486751005837405, −6.48417696131806974478363113845, −5.76021335812071984276534013575, −5.11799192727958313262232941974, −4.02790489914226635372002206787, −2.58219506594473837275894490570, −2.05380911314711136864921164756, 0,
2.05380911314711136864921164756, 2.58219506594473837275894490570, 4.02790489914226635372002206787, 5.11799192727958313262232941974, 5.76021335812071984276534013575, 6.48417696131806974478363113845, 7.68484370188773486751005837405, 8.289068147988507628565322456813, 9.134931013069261030986986227797
