| L(s) = 1 | + 3-s − 3·7-s + 9-s + 11-s + 4·19-s − 3·21-s + 6·23-s − 5·25-s + 27-s − 6·29-s + 5·31-s + 33-s − 6·37-s − 43-s − 4·47-s + 2·49-s + 6·53-s + 4·57-s + 14·59-s + 61-s − 3·63-s + 15·67-s + 6·69-s + 12·71-s − 15·73-s − 5·75-s − 3·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.917·19-s − 0.654·21-s + 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s + 0.898·31-s + 0.174·33-s − 0.986·37-s − 0.152·43-s − 0.583·47-s + 2/7·49-s + 0.824·53-s + 0.529·57-s + 1.82·59-s + 0.128·61-s − 0.377·63-s + 1.83·67-s + 0.722·69-s + 1.42·71-s − 1.75·73-s − 0.577·75-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.576893017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.576893017\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−13.21605100232790, −12.86106619510494, −12.24664877629161, −11.82875180928065, −11.30417499796615, −10.83139231787062, −10.10838504788047, −9.707260792702317, −9.536158525404919, −8.918281324683607, −8.418648070988512, −7.958704150032716, −7.262864733456838, −6.871756088298337, −6.599304243794962, −5.716454250102205, −5.442632588110531, −4.762616366937407, −3.988644324872205, −3.597122416336806, −3.162504229986606, −2.565321890263515, −1.938952403452053, −1.166840540077674, −0.4692216622226751,
0.4692216622226751, 1.166840540077674, 1.938952403452053, 2.565321890263515, 3.162504229986606, 3.597122416336806, 3.988644324872205, 4.762616366937407, 5.442632588110531, 5.716454250102205, 6.599304243794962, 6.871756088298337, 7.262864733456838, 7.958704150032716, 8.418648070988512, 8.918281324683607, 9.536158525404919, 9.707260792702317, 10.10838504788047, 10.83139231787062, 11.30417499796615, 11.82875180928065, 12.24664877629161, 12.86106619510494, 13.21605100232790
