| L(s) = 1 | − 7-s − 13-s − 7·17-s − 6·19-s + 2·23-s − 4·29-s − 8·31-s − 3·37-s − 6·41-s − 7·43-s − 3·47-s − 6·49-s − 6·53-s + 4·59-s + 10·61-s + 14·67-s + 71-s − 4·73-s − 16·79-s + 4·83-s + 2·89-s + 91-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.277·13-s − 1.69·17-s − 1.37·19-s + 0.417·23-s − 0.742·29-s − 1.43·31-s − 0.493·37-s − 0.937·41-s − 1.06·43-s − 0.437·47-s − 6/7·49-s − 0.824·53-s + 0.520·59-s + 1.28·61-s + 1.71·67-s + 0.118·71-s − 0.468·73-s − 1.80·79-s + 0.439·83-s + 0.211·89-s + 0.104·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−12.85249732763955, −12.44262422970424, −11.58466843154136, −11.41510126718100, −10.96954576273748, −10.47098140852374, −10.04326194533690, −9.488247820159611, −9.107669189376436, −8.533022657748527, −8.384322951870735, −7.660244443940034, −7.023319339578818, −6.698791975126680, −6.465723013676080, −5.706107890943358, −5.228015378176252, −4.737696516982941, −4.219083712140421, −3.698904364311261, −3.223609070120809, −2.529276068666263, −1.868013493137821, −1.746972614648907, −0.4925998876487965, 0,
0.4925998876487965, 1.746972614648907, 1.868013493137821, 2.529276068666263, 3.223609070120809, 3.698904364311261, 4.219083712140421, 4.737696516982941, 5.228015378176252, 5.706107890943358, 6.465723013676080, 6.698791975126680, 7.023319339578818, 7.660244443940034, 8.384322951870735, 8.533022657748527, 9.107669189376436, 9.488247820159611, 10.04326194533690, 10.47098140852374, 10.96954576273748, 11.41510126718100, 11.58466843154136, 12.44262422970424, 12.85249732763955
