| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 5·7-s + 8-s + 9-s + 10-s + 2·11-s − 12-s + 4·13-s + 5·14-s − 15-s + 16-s + 7·17-s + 18-s − 6·19-s + 20-s − 5·21-s + 2·22-s − 4·23-s − 24-s + 25-s + 4·26-s − 27-s + 5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 1.10·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s + 1.69·17-s + 0.235·18-s − 1.37·19-s + 0.223·20-s − 1.09·21-s + 0.426·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.843688476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.843688476\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 349 | \( 1 - T \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−16.77958498006453, −15.93605531420695, −15.30761868441699, −14.73176132669886, −14.19723240646447, −13.93931587148407, −13.16004167439243, −12.38135537861954, −11.99405704722581, −11.38290813864042, −10.89678594485477, −10.40434218783861, −9.700636015595254, −8.604983081449389, −8.290727607414656, −7.543362103077846, −6.802766083963137, −5.951000077439196, −5.648298283432340, −4.956867901391943, −4.159647718327588, −3.771048895864344, −2.445317071725533, −1.611317545659841, −1.098784594137887,
1.098784594137887, 1.611317545659841, 2.445317071725533, 3.771048895864344, 4.159647718327588, 4.956867901391943, 5.648298283432340, 5.951000077439196, 6.802766083963137, 7.543362103077846, 8.290727607414656, 8.604983081449389, 9.700636015595254, 10.40434218783861, 10.89678594485477, 11.38290813864042, 11.99405704722581, 12.38135537861954, 13.16004167439243, 13.93931587148407, 14.19723240646447, 14.73176132669886, 15.30761868441699, 15.93605531420695, 16.77958498006453
