| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s − 16-s + 8·17-s − 2·19-s − 20-s + 2·23-s + 25-s − 2·29-s + 2·31-s − 5·32-s − 8·34-s − 2·37-s + 2·38-s + 3·40-s − 2·41-s − 6·43-s − 2·46-s + 4·47-s − 50-s + 6·53-s + 2·58-s + 2·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s − 1/4·16-s + 1.94·17-s − 0.458·19-s − 0.223·20-s + 0.417·23-s + 1/5·25-s − 0.371·29-s + 0.359·31-s − 0.883·32-s − 1.37·34-s − 0.328·37-s + 0.324·38-s + 0.474·40-s − 0.312·41-s − 0.914·43-s − 0.294·46-s + 0.583·47-s − 0.141·50-s + 0.824·53-s + 0.262·58-s + 0.260·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.57804962754064, −12.36519419933969, −11.86870272512808, −11.23400180612181, −10.70574359731705, −10.36216131939935, −9.961408958379050, −9.571510038971301, −9.107514605174731, −8.737301608006042, −8.104157985300406, −7.827700663755983, −7.433322894049581, −6.665243010694654, −6.430107481823921, −5.519929881505498, −5.344286145615830, −4.889272229393878, −4.186795956960898, −3.634473709851728, −3.259774185584910, −2.450816810633794, −1.887425233848608, −1.199188128428162, −0.8403427978491756, 0,
0.8403427978491756, 1.199188128428162, 1.887425233848608, 2.450816810633794, 3.259774185584910, 3.634473709851728, 4.186795956960898, 4.889272229393878, 5.344286145615830, 5.519929881505498, 6.430107481823921, 6.665243010694654, 7.433322894049581, 7.827700663755983, 8.104157985300406, 8.737301608006042, 9.107514605174731, 9.571510038971301, 9.961408958379050, 10.36216131939935, 10.70574359731705, 11.23400180612181, 11.86870272512808, 12.36519419933969, 12.57804962754064
