| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 2·11-s − 12-s − 14-s − 15-s + 16-s − 4·17-s + 18-s + 19-s + 20-s + 21-s + 2·22-s + 3·23-s − 24-s + 25-s − 27-s − 28-s − 2·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s + 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s − 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.81149214971111, −14.12288131228237, −13.55608996601124, −13.42210772204692, −12.72444502539198, −12.16667226470982, −11.82340498542351, −11.29038893860837, −10.61268376353968, −10.35018520989604, −9.598623326359749, −9.143465383311594, −8.528061412492881, −7.835792798126441, −7.027732245143030, −6.597364022963134, −6.372648934715375, −5.554173565360913, −5.108592853790583, −4.496591311961644, −3.940008153049557, −3.186518425056200, −2.591797088756592, −1.752959772369243, −1.102319679142831, 0,
1.102319679142831, 1.752959772369243, 2.591797088756592, 3.186518425056200, 3.940008153049557, 4.496591311961644, 5.108592853790583, 5.554173565360913, 6.372648934715375, 6.597364022963134, 7.027732245143030, 7.835792798126441, 8.528061412492881, 9.143465383311594, 9.598623326359749, 10.35018520989604, 10.61268376353968, 11.29038893860837, 11.82340498542351, 12.16667226470982, 12.72444502539198, 13.42210772204692, 13.55608996601124, 14.12288131228237, 14.81149214971111
