| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 9-s + 6·11-s − 2·12-s + 2·13-s − 4·16-s + 2·17-s − 2·18-s + 4·19-s − 12·22-s − 6·23-s − 4·26-s − 27-s + 29-s + 5·31-s + 8·32-s − 6·33-s − 4·34-s + 2·36-s − 7·37-s − 8·38-s − 2·39-s − 10·41-s + 3·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 0.554·13-s − 16-s + 0.485·17-s − 0.471·18-s + 0.917·19-s − 2.55·22-s − 1.25·23-s − 0.784·26-s − 0.192·27-s + 0.185·29-s + 0.898·31-s + 1.41·32-s − 1.04·33-s − 0.685·34-s + 1/3·36-s − 1.15·37-s − 1.29·38-s − 0.320·39-s − 1.56·41-s + 0.457·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.295473999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.295473999\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−13.63307482972322, −13.42987672515655, −12.28579656051859, −12.01655819308499, −11.71937390656529, −11.27670464184269, −10.43978820299906, −10.27176293403353, −9.819804171229600, −9.106281098458286, −8.926770031279977, −8.315578785015601, −7.819117246287992, −7.129483714982977, −6.825100929540378, −6.306128318961734, −5.684050131836315, −5.159438811775049, −4.206032939265997, −3.973430400808312, −3.242561462648044, −2.244952664531982, −1.604347439839232, −1.059619525717298, −0.5731627768960467,
0.5731627768960467, 1.059619525717298, 1.604347439839232, 2.244952664531982, 3.242561462648044, 3.973430400808312, 4.206032939265997, 5.159438811775049, 5.684050131836315, 6.306128318961734, 6.825100929540378, 7.129483714982977, 7.819117246287992, 8.315578785015601, 8.926770031279977, 9.106281098458286, 9.819804171229600, 10.27176293403353, 10.43978820299906, 11.27670464184269, 11.71937390656529, 12.01655819308499, 12.28579656051859, 13.42987672515655, 13.63307482972322
