| L(s) = 1 | + 5-s + 3·7-s − 3·11-s + 13-s − 3·17-s − 4·19-s + 3·23-s + 25-s − 10·29-s + 6·31-s + 3·35-s − 5·37-s + 5·41-s − 2·43-s − 2·47-s + 2·49-s − 11·53-s − 3·55-s − 4·59-s + 61-s + 65-s + 4·67-s − 3·71-s + 6·73-s − 9·77-s − 3·79-s − 16·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s − 0.904·11-s + 0.277·13-s − 0.727·17-s − 0.917·19-s + 0.625·23-s + 1/5·25-s − 1.85·29-s + 1.07·31-s + 0.507·35-s − 0.821·37-s + 0.780·41-s − 0.304·43-s − 0.291·47-s + 2/7·49-s − 1.51·53-s − 0.404·55-s − 0.520·59-s + 0.128·61-s + 0.124·65-s + 0.488·67-s − 0.356·71-s + 0.702·73-s − 1.02·77-s − 0.337·79-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−7.45194530323652218418946645979, −6.66855499581163136787937930648, −5.96119896044861983869100043753, −5.20607498160861371439791450887, −4.72244886329515518619099132946, −3.94731752547646479370237626667, −2.88535063519522669732758213725, −2.10342140054844298671384284426, −1.42362098811623626262880975902, 0,
1.42362098811623626262880975902, 2.10342140054844298671384284426, 2.88535063519522669732758213725, 3.94731752547646479370237626667, 4.72244886329515518619099132946, 5.20607498160861371439791450887, 5.96119896044861983869100043753, 6.66855499581163136787937930648, 7.45194530323652218418946645979
