| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 4·11-s + 12-s − 15-s + 16-s − 18-s − 2·19-s − 20-s − 4·22-s − 4·23-s − 24-s + 25-s + 27-s + 2·29-s + 30-s + 6·31-s − 32-s + 4·33-s + 36-s + 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.182·30-s + 1.07·31-s − 0.176·32-s + 0.696·33-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.17216931525286, −12.39752296644618, −12.10237954486123, −11.71806603800074, −11.30750946563983, −10.60440303223458, −10.29660228715831, −9.722878049496351, −9.392056417884484, −8.790268648948303, −8.420241931735379, −8.127850939002356, −7.496235205436550, −7.050532682089233, −6.466612588088467, −6.270946231229695, −5.478365019307403, −4.790796805510307, −4.196584435689605, −3.794308260904494, −3.303314218858726, −2.510295530736198, −2.159625732825076, −1.329886073447998, −0.8948227119866757, 0,
0.8948227119866757, 1.329886073447998, 2.159625732825076, 2.510295530736198, 3.303314218858726, 3.794308260904494, 4.196584435689605, 4.790796805510307, 5.478365019307403, 6.270946231229695, 6.466612588088467, 7.050532682089233, 7.496235205436550, 8.127850939002356, 8.420241931735379, 8.790268648948303, 9.392056417884484, 9.722878049496351, 10.29660228715831, 10.60440303223458, 11.30750946563983, 11.71806603800074, 12.10237954486123, 12.39752296644618, 13.17216931525286
