| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 3·7-s − 8-s + 9-s − 10-s − 3·11-s + 12-s − 13-s − 3·14-s + 15-s + 16-s + 4·17-s − 18-s + 2·19-s + 20-s + 3·21-s + 3·22-s − 24-s + 25-s + 26-s + 27-s + 3·28-s + 9·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.654·21-s + 0.639·22-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.566·28-s + 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.454182209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.454182209\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 23 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−13.01512974229110, −12.51856596445830, −12.01695273118114, −11.63646831161687, −11.07997722299959, −10.39936972374959, −10.28651499443128, −9.793572549521760, −9.254032252948216, −8.673075759762059, −8.256901383633345, −7.923877239252300, −7.489479853792994, −7.027641575097171, −6.305386108278765, −5.819394083359416, −5.206014918918862, −4.739124520740706, −4.314002233184685, −3.337971631493055, −2.808454448239492, −2.519343768502599, −1.661981575370384, −1.287823666281047, −0.5812941439638796,
0.5812941439638796, 1.287823666281047, 1.661981575370384, 2.519343768502599, 2.808454448239492, 3.337971631493055, 4.314002233184685, 4.739124520740706, 5.206014918918862, 5.819394083359416, 6.305386108278765, 7.027641575097171, 7.489479853792994, 7.923877239252300, 8.256901383633345, 8.673075759762059, 9.254032252948216, 9.793572549521760, 10.28651499443128, 10.39936972374959, 11.07997722299959, 11.63646831161687, 12.01695273118114, 12.51856596445830, 13.01512974229110
