| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 4·11-s + 12-s − 3·13-s − 14-s + 15-s + 16-s + 7·17-s − 18-s + 5·19-s + 20-s + 21-s + 4·22-s − 3·23-s − 24-s + 25-s + 3·26-s + 27-s + 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 + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.588·26-s + 0.192·27-s + 0.188·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 349 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 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
−16.89008920898205, −16.23698307246447, −15.86946902851427, −15.17542815121005, −14.48886815396489, −14.16689470731400, −13.51494593308240, −12.77341819945965, −12.19880421328505, −11.69935515994864, −10.81480601964059, −10.21000240237547, −9.778154832617034, −9.425203258284357, −8.325743005289800, −8.066837710841140, −7.482932878597471, −6.894116869589340, −5.895383169440437, −5.230217576540750, −4.751969679139291, −3.268286934063002, −3.077708895681517, −1.982700696129052, −1.369205244007557, 0,
1.369205244007557, 1.982700696129052, 3.077708895681517, 3.268286934063002, 4.751969679139291, 5.230217576540750, 5.895383169440437, 6.894116869589340, 7.482932878597471, 8.066837710841140, 8.325743005289800, 9.425203258284357, 9.778154832617034, 10.21000240237547, 10.81480601964059, 11.69935515994864, 12.19880421328505, 12.77341819945965, 13.51494593308240, 14.16689470731400, 14.48886815396489, 15.17542815121005, 15.86946902851427, 16.23698307246447, 16.89008920898205
