| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 4·11-s + 4·13-s + 14-s + 16-s + 17-s + 20-s + 4·22-s + 2·23-s + 25-s − 4·26-s − 28-s + 2·29-s + 6·31-s − 32-s − 34-s − 35-s − 10·37-s − 40-s − 10·41-s − 4·44-s − 2·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.223·20-s + 0.852·22-s + 0.417·23-s + 1/5·25-s − 0.784·26-s − 0.188·28-s + 0.371·29-s + 1.07·31-s − 0.176·32-s − 0.171·34-s − 0.169·35-s − 1.64·37-s − 0.158·40-s − 1.56·41-s − 0.603·44-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−17.00401416453718, −16.11527849118891, −15.78102651698834, −15.46118932312356, −14.55252320760671, −13.91689032878118, −13.28882314066649, −12.94469275421688, −12.14197357375821, −11.52960982054790, −10.83717252158790, −10.19621335801680, −10.06309552384608, −9.090676471827634, −8.548414363974352, −8.104693138567405, −7.273431441629327, −6.643575942221869, −6.036232913945689, −5.347492806547032, −4.650152952118339, −3.437503496747223, −2.979391281450705, −2.012482661631330, −1.162636981249582, 0,
1.162636981249582, 2.012482661631330, 2.979391281450705, 3.437503496747223, 4.650152952118339, 5.347492806547032, 6.036232913945689, 6.643575942221869, 7.273431441629327, 8.104693138567405, 8.548414363974352, 9.090676471827634, 10.06309552384608, 10.19621335801680, 10.83717252158790, 11.52960982054790, 12.14197357375821, 12.94469275421688, 13.28882314066649, 13.91689032878118, 14.55252320760671, 15.46118932312356, 15.78102651698834, 16.11527849118891, 17.00401416453718
