| L(s) = 1 | + 2·5-s − 7-s − 3·9-s + 2·11-s − 17-s + 2·19-s − 8·23-s − 25-s + 8·31-s − 2·35-s + 4·37-s − 6·41-s − 4·43-s − 6·45-s + 8·47-s + 49-s + 6·53-s + 4·55-s − 10·59-s − 10·61-s + 3·63-s − 8·67-s + 4·71-s − 10·73-s − 2·77-s − 4·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s + 0.603·11-s − 0.242·17-s + 0.458·19-s − 1.66·23-s − 1/5·25-s + 1.43·31-s − 0.338·35-s + 0.657·37-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 + 0.539·55-s − 1.30·59-s − 1.28·61-s + 0.377·63-s − 0.977·67-s + 0.474·71-s − 1.17·73-s − 0.227·77-s − 0.450·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 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 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−7.58065768620980265753202168904, −6.61004184088692613218328726141, −6.05072788211209209834975959834, −5.71154266238812760623566557968, −4.72091276724513618684329234140, −3.90689476429918404644061307931, −2.99882986699961188322134332297, −2.30468846520901319372603625349, −1.35825535308942368395974391982, 0,
1.35825535308942368395974391982, 2.30468846520901319372603625349, 2.99882986699961188322134332297, 3.90689476429918404644061307931, 4.72091276724513618684329234140, 5.71154266238812760623566557968, 6.05072788211209209834975959834, 6.61004184088692613218328726141, 7.58065768620980265753202168904
