| L(s) = 1 | + 3-s + 9-s − 6·11-s − 4·13-s + 19-s + 6·23-s − 5·25-s + 27-s − 6·29-s + 4·31-s − 6·33-s − 2·37-s − 4·39-s + 6·41-s − 8·43-s − 12·47-s − 6·53-s + 57-s − 12·59-s − 10·61-s − 14·67-s + 6·69-s − 12·71-s − 2·73-s − 5·75-s − 10·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.229·19-s + 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 1.04·33-s − 0.328·37-s − 0.640·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s − 0.824·53-s + 0.132·57-s − 1.56·59-s − 1.28·61-s − 1.71·67-s + 0.722·69-s − 1.42·71-s − 0.234·73-s − 0.577·75-s − 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.57710258309829, −13.12878644826374, −12.88277730872095, −12.30684732748986, −11.81755541715315, −11.18826301946653, −10.81843724200260, −10.19698697076939, −9.874926239205441, −9.434062176887033, −8.880792801264773, −8.320935401031106, −7.758582651554765, −7.508802541510614, −7.154620922039678, −6.314130298889939, −5.812953229986304, −5.185320781112643, −4.753693420333139, −4.414705336172268, −3.407022287571265, −2.989335801728863, −2.656702711140368, −1.883402327512315, −1.379520659757447, 0, 0,
1.379520659757447, 1.883402327512315, 2.656702711140368, 2.989335801728863, 3.407022287571265, 4.414705336172268, 4.753693420333139, 5.185320781112643, 5.812953229986304, 6.314130298889939, 7.154620922039678, 7.508802541510614, 7.758582651554765, 8.320935401031106, 8.880792801264773, 9.434062176887033, 9.874926239205441, 10.19698697076939, 10.81843724200260, 11.18826301946653, 11.81755541715315, 12.30684732748986, 12.88277730872095, 13.12878644826374, 13.57710258309829
