| L(s) = 1 | + 5-s − 2·7-s + 4·11-s − 2·13-s + 19-s − 8·23-s + 25-s + 2·29-s + 10·31-s − 2·35-s + 6·37-s − 12·41-s + 8·43-s − 8·47-s − 3·49-s + 6·53-s + 4·55-s + 6·59-s + 10·61-s − 2·65-s − 4·67-s − 2·73-s − 8·77-s + 10·79-s + 6·83-s + 4·91-s + 95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 1.20·11-s − 0.554·13-s + 0.229·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s + 1.79·31-s − 0.338·35-s + 0.986·37-s − 1.87·41-s + 1.21·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.539·55-s + 0.781·59-s + 1.28·61-s − 0.248·65-s − 0.488·67-s − 0.234·73-s − 0.911·77-s + 1.12·79-s + 0.658·83-s + 0.419·91-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.982342554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.982342554\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.053984250624830108949386248547, −7.11860656391730210479111292980, −6.40915918060034412867450797096, −6.13959055082787592841858036672, −5.13228821731546196103126441907, −4.31650556846324797817852036847, −3.60249543103238411798979282877, −2.71741286669436281375339226070, −1.85107869847392145428320374516, −0.72140743868074912833281881137,
0.72140743868074912833281881137, 1.85107869847392145428320374516, 2.71741286669436281375339226070, 3.60249543103238411798979282877, 4.31650556846324797817852036847, 5.13228821731546196103126441907, 6.13959055082787592841858036672, 6.40915918060034412867450797096, 7.11860656391730210479111292980, 8.053984250624830108949386248547
