| L(s) = 1 | + 5-s − 3·9-s + 4·13-s − 4·17-s + 8·19-s + 4·23-s + 25-s + 8·29-s + 4·31-s + 6·37-s − 8·41-s − 4·43-s − 3·45-s − 12·47-s − 10·53-s + 8·61-s + 4·65-s + 8·67-s + 12·71-s + 12·73-s + 8·79-s + 9·81-s + 4·83-s − 4·85-s − 10·89-s + 8·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s + 1.10·13-s − 0.970·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s + 1.48·29-s + 0.718·31-s + 0.986·37-s − 1.24·41-s − 0.609·43-s − 0.447·45-s − 1.75·47-s − 1.37·53-s + 1.02·61-s + 0.496·65-s + 0.977·67-s + 1.42·71-s + 1.40·73-s + 0.900·79-s + 81-s + 0.439·83-s − 0.433·85-s − 1.05·89-s + 0.820·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118580 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.254383036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.254383036\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 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.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.67384282170576, −13.16213089222351, −12.75588813402181, −11.95571144611262, −11.63052905202932, −11.05303800975158, −10.96505880085922, −9.966428409766592, −9.772079492230900, −9.157131516248804, −8.627104579877141, −8.188898203259111, −7.866273797997961, −6.828040948063967, −6.583312509768420, −6.176190570334232, −5.390497166186138, −5.035085550688703, −4.570699209565358, −3.511533052751634, −3.275394343882204, −2.658805600982326, −1.953888245086521, −1.133763435751796, −0.6185856735562790,
0.6185856735562790, 1.133763435751796, 1.953888245086521, 2.658805600982326, 3.275394343882204, 3.511533052751634, 4.570699209565358, 5.035085550688703, 5.390497166186138, 6.176190570334232, 6.583312509768420, 6.828040948063967, 7.866273797997961, 8.188898203259111, 8.627104579877141, 9.157131516248804, 9.772079492230900, 9.966428409766592, 10.96505880085922, 11.05303800975158, 11.63052905202932, 11.95571144611262, 12.75588813402181, 13.16213089222351, 13.67384282170576
