| L(s) = 1 | + 5-s − 2·7-s + 4·11-s + 2·13-s − 6·17-s + 3·19-s + 6·23-s − 4·25-s + 6·29-s − 11·31-s − 2·35-s + 6·37-s − 2·41-s + 4·43-s − 2·47-s − 3·49-s + 11·53-s + 4·55-s + 3·59-s − 6·61-s + 2·65-s − 5·67-s + 16·71-s − 3·73-s − 8·77-s + 9·79-s + 6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 1.20·11-s + 0.554·13-s − 1.45·17-s + 0.688·19-s + 1.25·23-s − 4/5·25-s + 1.11·29-s − 1.97·31-s − 0.338·35-s + 0.986·37-s − 0.312·41-s + 0.609·43-s − 0.291·47-s − 3/7·49-s + 1.51·53-s + 0.539·55-s + 0.390·59-s − 0.768·61-s + 0.248·65-s − 0.610·67-s + 1.89·71-s − 0.351·73-s − 0.911·77-s + 1.01·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.043361022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.043361022\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 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.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.10246564750539, −12.50757056118560, −12.05696003197230, −11.32194662621795, −11.25936187374505, −10.66989791622210, −10.04817362590014, −9.569850291624833, −9.164483432357183, −8.898107773782124, −8.413547934619303, −7.577605956381895, −7.172961178116187, −6.626995662428728, −6.305588808750175, −5.868059439024846, −5.218801836394785, −4.656669680334513, −4.062113753247681, −3.552017673466343, −3.132279571308820, −2.318010770525974, −1.879903239150786, −1.089844513132939, −0.5312950533317577,
0.5312950533317577, 1.089844513132939, 1.879903239150786, 2.318010770525974, 3.132279571308820, 3.552017673466343, 4.062113753247681, 4.656669680334513, 5.218801836394785, 5.868059439024846, 6.305588808750175, 6.626995662428728, 7.172961178116187, 7.577605956381895, 8.413547934619303, 8.898107773782124, 9.164483432357183, 9.569850291624833, 10.04817362590014, 10.66989791622210, 11.25936187374505, 11.32194662621795, 12.05696003197230, 12.50757056118560, 13.10246564750539
