| L(s) = 1 | − 3·7-s + 11-s − 5·13-s − 6·17-s − 19-s − 6·23-s − 6·29-s − 5·31-s − 6·37-s − 10·41-s − 7·43-s − 4·47-s + 2·49-s − 2·53-s − 12·59-s + 9·61-s − 13·67-s − 2·73-s − 3·77-s + 12·79-s + 2·83-s + 6·89-s + 15·91-s + 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.301·11-s − 1.38·13-s − 1.45·17-s − 0.229·19-s − 1.25·23-s − 1.11·29-s − 0.898·31-s − 0.986·37-s − 1.56·41-s − 1.06·43-s − 0.583·47-s + 2/7·49-s − 0.274·53-s − 1.56·59-s + 1.15·61-s − 1.58·67-s − 0.234·73-s − 0.341·77-s + 1.35·79-s + 0.219·83-s + 0.635·89-s + 1.57·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−16.31976677310282, −15.56973165127116, −15.22052496565869, −14.65891808973051, −13.99461005318605, −13.43150960598156, −12.93198937681869, −12.45820503842882, −11.85015329296679, −11.37512470999073, −10.56415279948520, −10.01920354764964, −9.631177279338753, −8.952686422146794, −8.532433828036356, −7.503982117488293, −7.203185808377352, −6.377658137851227, −6.141497180488928, −5.067996947307905, −4.668080870499089, −3.671573826292042, −3.316003802297628, −2.214787304566340, −1.833583885083313, 0, 0,
1.833583885083313, 2.214787304566340, 3.316003802297628, 3.671573826292042, 4.668080870499089, 5.067996947307905, 6.141497180488928, 6.377658137851227, 7.203185808377352, 7.503982117488293, 8.532433828036356, 8.952686422146794, 9.631177279338753, 10.01920354764964, 10.56415279948520, 11.37512470999073, 11.85015329296679, 12.45820503842882, 12.93198937681869, 13.43150960598156, 13.99461005318605, 14.65891808973051, 15.22052496565869, 15.56973165127116, 16.31976677310282
