| L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·11-s − 13-s + 2·15-s + 2·17-s + 4·19-s + 4·23-s − 25-s − 27-s + 8·29-s − 4·31-s + 4·33-s + 4·37-s + 39-s − 2·45-s + 6·47-s − 2·51-s − 12·53-s + 8·55-s − 4·57-s − 2·61-s + 2·65-s + 2·67-s − 4·69-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.516·15-s + 0.485·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.696·33-s + 0.657·37-s + 0.160·39-s − 0.298·45-s + 0.875·47-s − 0.280·51-s − 1.64·53-s + 1.07·55-s − 0.529·57-s − 0.256·61-s + 0.248·65-s + 0.244·67-s − 0.481·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.79299020288691, −13.23744461957571, −12.63531823980518, −12.33173384177576, −11.94671679336272, −11.22787850739131, −11.05447360506597, −10.50184776403023, −9.905436973341818, −9.554763166197478, −8.878076590281586, −8.172928490051406, −7.823998616174901, −7.459606052882276, −6.887041978237669, −6.329792924508531, −5.591871827702068, −5.226198086866741, −4.732374913996492, −4.173479244705550, −3.461811075995652, −2.944589345111901, −2.385545083215611, −1.380560923336144, −0.7239684627914787, 0,
0.7239684627914787, 1.380560923336144, 2.385545083215611, 2.944589345111901, 3.461811075995652, 4.173479244705550, 4.732374913996492, 5.226198086866741, 5.591871827702068, 6.329792924508531, 6.887041978237669, 7.459606052882276, 7.823998616174901, 8.172928490051406, 8.878076590281586, 9.554763166197478, 9.905436973341818, 10.50184776403023, 11.05447360506597, 11.22787850739131, 11.94671679336272, 12.33173384177576, 12.63531823980518, 13.23744461957571, 13.79299020288691
