| L(s) = 1 | − 5-s − 3·9-s − 5·11-s − 13-s − 6·17-s − 4·19-s − 4·25-s − 5·31-s + 37-s − 2·41-s − 2·43-s + 3·45-s + 2·47-s − 9·53-s + 5·55-s + 3·59-s + 6·61-s + 65-s − 5·67-s + 71-s − 10·73-s + 4·79-s + 9·81-s + 14·83-s + 6·85-s − 15·89-s + 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 1.50·11-s − 0.277·13-s − 1.45·17-s − 0.917·19-s − 4/5·25-s − 0.898·31-s + 0.164·37-s − 0.312·41-s − 0.304·43-s + 0.447·45-s + 0.291·47-s − 1.23·53-s + 0.674·55-s + 0.390·59-s + 0.768·61-s + 0.124·65-s − 0.610·67-s + 0.118·71-s − 1.17·73-s + 0.450·79-s + 81-s + 1.53·83-s + 0.650·85-s − 1.58·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.85200327208217, −13.24633694233743, −12.92072543127137, −12.51691226328146, −11.79579834266415, −11.26920960910099, −11.13104495004938, −10.39410285430924, −10.16293688713570, −9.288788776649815, −8.882437220572282, −8.413130347371378, −7.895195576154795, −7.553646286226994, −6.836712253458074, −6.333380251552055, −5.719601678192633, −5.270666497660024, −4.648565128731928, −4.199871144391558, −3.426473646346900, −2.889893853030241, −2.220847868391351, −1.898675389038013, −0.4818783795034909, 0,
0.4818783795034909, 1.898675389038013, 2.220847868391351, 2.889893853030241, 3.426473646346900, 4.199871144391558, 4.648565128731928, 5.270666497660024, 5.719601678192633, 6.333380251552055, 6.836712253458074, 7.553646286226994, 7.895195576154795, 8.413130347371378, 8.882437220572282, 9.288788776649815, 10.16293688713570, 10.39410285430924, 11.13104495004938, 11.26920960910099, 11.79579834266415, 12.51691226328146, 12.92072543127137, 13.24633694233743, 13.85200327208217
