| L(s) = 1 | + 2-s + 4-s − 2·5-s − 2·7-s + 8-s − 2·10-s + 4·11-s + 4·13-s − 2·14-s + 16-s − 2·17-s − 2·20-s + 4·22-s − 25-s + 4·26-s − 2·28-s + 4·31-s + 32-s − 2·34-s + 4·35-s − 2·37-s − 2·40-s + 41-s + 12·43-s + 4·44-s + 2·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s + 0.353·8-s − 0.632·10-s + 1.20·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.784·26-s − 0.377·28-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 0.676·35-s − 0.328·37-s − 0.316·40-s + 0.156·41-s + 1.82·43-s + 0.603·44-s + 0.291·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−12.66647127985728, −12.27482446748712, −11.77496638506031, −11.43331145730024, −11.03140884451622, −10.62010828049044, −10.02648282404805, −9.414716275161395, −9.107671242431335, −8.585389308907213, −8.075016615770629, −7.577794304687541, −7.120338717146647, −6.443737374013718, −6.332403395960556, −5.902937620113111, −5.128186827276616, −4.608820061844519, −4.016288289122487, −3.755923173454143, −3.428691711717183, −2.708003020894426, −2.157465578842647, −1.324884823754281, −0.8583744250087138, 0,
0.8583744250087138, 1.324884823754281, 2.157465578842647, 2.708003020894426, 3.428691711717183, 3.755923173454143, 4.016288289122487, 4.608820061844519, 5.128186827276616, 5.902937620113111, 6.332403395960556, 6.443737374013718, 7.120338717146647, 7.577794304687541, 8.075016615770629, 8.585389308907213, 9.107671242431335, 9.414716275161395, 10.02648282404805, 10.62010828049044, 11.03140884451622, 11.43331145730024, 11.77496638506031, 12.27482446748712, 12.66647127985728
