| L(s) = 1 | + 3-s + 9-s − 5·11-s + 2·13-s − 8·17-s − 19-s + 23-s − 5·25-s + 27-s + 6·29-s − 11·31-s − 5·33-s + 4·37-s + 2·39-s + 41-s + 9·43-s + 47-s − 8·51-s − 10·53-s − 57-s − 3·59-s + 3·61-s − 8·67-s + 69-s + 10·71-s + 4·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.554·13-s − 1.94·17-s − 0.229·19-s + 0.208·23-s − 25-s + 0.192·27-s + 1.11·29-s − 1.97·31-s − 0.870·33-s + 0.657·37-s + 0.320·39-s + 0.156·41-s + 1.37·43-s + 0.145·47-s − 1.12·51-s − 1.37·53-s − 0.132·57-s − 0.390·59-s + 0.384·61-s − 0.977·67-s + 0.120·69-s + 1.18·71-s + 0.468·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−12.82072035714384, −12.49456128901689, −11.69063184842769, −11.17407873575134, −10.83237730702414, −10.64195930919132, −9.871615422759795, −9.548176198878909, −8.944873709546731, −8.662462385424882, −8.142070784020530, −7.734231134201146, −7.239157877556067, −6.834786871678594, −6.137247421764516, −5.811632826790529, −5.223203756114325, −4.528536339533197, −4.334149410557163, −3.681426560051352, −3.027450346756683, −2.592593136603160, −2.070517080930411, −1.640606225240246, −0.6320649687659124, 0,
0.6320649687659124, 1.640606225240246, 2.070517080930411, 2.592593136603160, 3.027450346756683, 3.681426560051352, 4.334149410557163, 4.528536339533197, 5.223203756114325, 5.811632826790529, 6.137247421764516, 6.834786871678594, 7.239157877556067, 7.734231134201146, 8.142070784020530, 8.662462385424882, 8.944873709546731, 9.548176198878909, 9.871615422759795, 10.64195930919132, 10.83237730702414, 11.17407873575134, 11.69063184842769, 12.49456128901689, 12.82072035714384
