| L(s) = 1 | − 3-s − 2·5-s + 9-s − 3·11-s − 2·13-s + 2·15-s + 17-s − 19-s + 7·23-s − 25-s − 27-s − 10·29-s + 6·31-s + 3·33-s + 8·37-s + 2·39-s + 6·41-s − 4·43-s − 2·45-s + 9·47-s − 51-s + 4·53-s + 6·55-s + 57-s + 6·59-s + 61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.904·11-s − 0.554·13-s + 0.516·15-s + 0.242·17-s − 0.229·19-s + 1.45·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s + 1.07·31-s + 0.522·33-s + 1.31·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s + 1.31·47-s − 0.140·51-s + 0.549·53-s + 0.809·55-s + 0.132·57-s + 0.781·59-s + 0.128·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.23665108580241, −12.78776360876739, −12.60859248389369, −11.87271670016354, −11.41155692030966, −11.25582160955595, −10.63594465874427, −10.17017005101979, −9.672697441109582, −9.168151120801438, −8.565309639520468, −7.998733621159883, −7.598050319409732, −7.181947642893920, −6.782657524234204, −5.928222954363392, −5.568196559025366, −5.109283175742751, −4.395465108187963, −4.134530810896520, −3.421744699013263, −2.712706347016029, −2.336639582281655, −1.337238048338776, −0.6582026517026862, 0,
0.6582026517026862, 1.337238048338776, 2.336639582281655, 2.712706347016029, 3.421744699013263, 4.134530810896520, 4.395465108187963, 5.109283175742751, 5.568196559025366, 5.928222954363392, 6.782657524234204, 7.181947642893920, 7.598050319409732, 7.998733621159883, 8.565309639520468, 9.168151120801438, 9.672697441109582, 10.17017005101979, 10.63594465874427, 11.25582160955595, 11.41155692030966, 11.87271670016354, 12.60859248389369, 12.78776360876739, 13.23665108580241
