| L(s) = 1 | − 2·5-s + 13-s − 2·17-s − 4·19-s − 25-s − 6·29-s − 2·37-s − 6·41-s − 12·43-s + 4·47-s − 7·49-s − 6·53-s + 8·59-s − 2·61-s − 2·65-s + 4·67-s + 12·71-s − 14·73-s − 8·83-s + 4·85-s + 18·89-s + 8·95-s − 6·97-s − 14·101-s + 16·103-s − 4·107-s − 2·109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.277·13-s − 0.485·17-s − 0.917·19-s − 1/5·25-s − 1.11·29-s − 0.328·37-s − 0.937·41-s − 1.82·43-s + 0.583·47-s − 49-s − 0.824·53-s + 1.04·59-s − 0.256·61-s − 0.248·65-s + 0.488·67-s + 1.42·71-s − 1.63·73-s − 0.878·83-s + 0.433·85-s + 1.90·89-s + 0.820·95-s − 0.609·97-s − 1.39·101-s + 1.57·103-s − 0.386·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.637423071689071752397002310776, −8.640567914893326663493991937073, −8.067557263920253412091787940410, −7.11188121211474152789224421277, −6.31368244093400769096099024522, −5.15302223585186903893080651929, −4.15951728328837579927312423983, −3.35188513577727550298605037893, −1.88987547858298345510585245863, 0,
1.88987547858298345510585245863, 3.35188513577727550298605037893, 4.15951728328837579927312423983, 5.15302223585186903893080651929, 6.31368244093400769096099024522, 7.11188121211474152789224421277, 8.067557263920253412091787940410, 8.640567914893326663493991937073, 9.637423071689071752397002310776
