| L(s) = 1 | − 3-s − 2·4-s + 5-s − 2·7-s + 9-s − 5·11-s + 2·12-s + 5·13-s − 15-s + 4·16-s + 5·17-s − 2·20-s + 2·21-s − 9·23-s + 25-s − 27-s + 4·28-s − 8·29-s + 5·31-s + 5·33-s − 2·35-s − 2·36-s − 8·37-s − 5·39-s + 7·41-s − 43-s + 10·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.50·11-s + 0.577·12-s + 1.38·13-s − 0.258·15-s + 16-s + 1.21·17-s − 0.447·20-s + 0.436·21-s − 1.87·23-s + 1/5·25-s − 0.192·27-s + 0.755·28-s − 1.48·29-s + 0.898·31-s + 0.870·33-s − 0.338·35-s − 1/3·36-s − 1.31·37-s − 0.800·39-s + 1.09·41-s − 0.152·43-s + 1.50·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| 43 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−13.15473161068632, −12.71025037728159, −12.46614369024373, −11.75564044390747, −11.33838262537567, −10.62955058505075, −10.18001940514679, −10.05576223299790, −9.594150449652743, −8.938667141276163, −8.437581693009679, −8.031022358549612, −7.562194694077543, −6.993419594998378, −6.145611660267139, −5.768902809239716, −5.722608361622738, −5.025769143501344, −4.423707445346129, −3.854021393101853, −3.353294216662817, −2.914357046349138, −1.947553081201195, −1.420529377544086, −0.5740596958117184, 0,
0.5740596958117184, 1.420529377544086, 1.947553081201195, 2.914357046349138, 3.353294216662817, 3.854021393101853, 4.423707445346129, 5.025769143501344, 5.722608361622738, 5.768902809239716, 6.145611660267139, 6.993419594998378, 7.562194694077543, 8.031022358549612, 8.437581693009679, 8.938667141276163, 9.594150449652743, 10.05576223299790, 10.18001940514679, 10.62955058505075, 11.33838262537567, 11.75564044390747, 12.46614369024373, 12.71025037728159, 13.15473161068632
