| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 13-s + 15-s + 5·17-s + 8·19-s − 2·21-s − 6·23-s − 4·25-s − 27-s + 3·29-s − 4·31-s − 2·35-s − 11·37-s − 39-s − 3·41-s − 2·43-s − 45-s − 4·47-s − 3·49-s − 5·51-s + 3·53-s − 8·57-s − 6·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 1.21·17-s + 1.83·19-s − 0.436·21-s − 1.25·23-s − 4/5·25-s − 0.192·27-s + 0.557·29-s − 0.718·31-s − 0.338·35-s − 1.80·37-s − 0.160·39-s − 0.468·41-s − 0.304·43-s − 0.149·45-s − 0.583·47-s − 3/7·49-s − 0.700·51-s + 0.412·53-s − 1.05·57-s − 0.781·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−16.52493960027732, −16.20418131975348, −15.69350045549844, −15.08002847428448, −14.37679947529626, −13.84893960782482, −13.48497330803452, −12.34267494329890, −12.05053884837373, −11.69704423854611, −11.02480680642124, −10.34453927037341, −9.854182347170728, −9.178740375805269, −8.266352468336179, −7.797907247772826, −7.358133910493936, −6.513885789687253, −5.672430043821636, −5.288484518492506, −4.580874278065912, −3.664952428964020, −3.210082905121656, −1.838840865423536, −1.222780200252140, 0,
1.222780200252140, 1.838840865423536, 3.210082905121656, 3.664952428964020, 4.580874278065912, 5.288484518492506, 5.672430043821636, 6.513885789687253, 7.358133910493936, 7.797907247772826, 8.266352468336179, 9.178740375805269, 9.854182347170728, 10.34453927037341, 11.02480680642124, 11.69704423854611, 12.05053884837373, 12.34267494329890, 13.48497330803452, 13.84893960782482, 14.37679947529626, 15.08002847428448, 15.69350045549844, 16.20418131975348, 16.52493960027732
