| L(s) = 1 | − 5-s − 5·11-s − 2·13-s − 5·17-s − 8·23-s + 25-s + 4·31-s − 2·37-s + 41-s + 10·43-s + 3·47-s − 7·49-s − 2·53-s + 5·55-s − 15·59-s + 7·61-s + 2·65-s − 67-s + 2·71-s + 14·73-s − 6·79-s + 17·83-s + 5·85-s + 89-s − 17·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.50·11-s − 0.554·13-s − 1.21·17-s − 1.66·23-s + 1/5·25-s + 0.718·31-s − 0.328·37-s + 0.156·41-s + 1.52·43-s + 0.437·47-s − 49-s − 0.274·53-s + 0.674·55-s − 1.95·59-s + 0.896·61-s + 0.248·65-s − 0.122·67-s + 0.237·71-s + 1.63·73-s − 0.675·79-s + 1.86·83-s + 0.542·85-s + 0.105·89-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 131 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−14.06398447111861, −13.57185516620169, −13.07111074218286, −12.50474564983134, −12.23540160020589, −11.62592467827579, −10.91793262804932, −10.79477719082544, −10.10981840063541, −9.663640439212222, −9.110693629353257, −8.382277853166601, −8.073201107362931, −7.553108835260224, −7.151339429201858, −6.278762711722724, −6.054903009440708, −5.166866960672461, −4.802688086246628, −4.234752852362407, −3.638896021165777, −2.826244604735413, −2.370189191657280, −1.824945263500838, −0.6286853703870724, 0,
0.6286853703870724, 1.824945263500838, 2.370189191657280, 2.826244604735413, 3.638896021165777, 4.234752852362407, 4.802688086246628, 5.166866960672461, 6.054903009440708, 6.278762711722724, 7.151339429201858, 7.553108835260224, 8.073201107362931, 8.382277853166601, 9.110693629353257, 9.663640439212222, 10.10981840063541, 10.79477719082544, 10.91793262804932, 11.62592467827579, 12.23540160020589, 12.50474564983134, 13.07111074218286, 13.57185516620169, 14.06398447111861
