| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s − 12-s − 13-s + 14-s + 2·15-s + 16-s − 8·17-s − 18-s − 4·19-s − 2·20-s + 21-s + 6·23-s + 24-s − 25-s + 26-s − 27-s − 28-s − 2·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s + 1.25·23-s + 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−14.74612951694068, −13.96481355443585, −13.19396261937077, −12.75931037786050, −12.57833663170517, −11.67330027637259, −11.33992074944393, −10.94928747459798, −10.61964171047150, −9.813800638127149, −9.352338195937759, −8.748361780963320, −8.488219374193012, −7.571742858845555, −7.267032049893030, −6.815170386712557, −6.102975767630579, −5.785420198465841, −4.686754554975100, −4.437037368613621, −3.804925073618590, −2.933595527270659, −2.362225894313467, −1.580203220453143, −0.5799640736343976, 0,
0.5799640736343976, 1.580203220453143, 2.362225894313467, 2.933595527270659, 3.804925073618590, 4.437037368613621, 4.686754554975100, 5.785420198465841, 6.102975767630579, 6.815170386712557, 7.267032049893030, 7.571742858845555, 8.488219374193012, 8.748361780963320, 9.352338195937759, 9.813800638127149, 10.61964171047150, 10.94928747459798, 11.33992074944393, 11.67330027637259, 12.57833663170517, 12.75931037786050, 13.19396261937077, 13.96481355443585, 14.74612951694068
