| L(s) = 1 | − 5-s + 4·11-s + 6·13-s − 2·17-s + 8·19-s − 4·23-s + 25-s − 6·29-s − 4·31-s − 2·37-s − 2·41-s − 12·43-s − 2·53-s − 4·55-s − 4·59-s − 6·61-s − 6·65-s − 4·67-s − 8·71-s − 6·73-s − 16·79-s − 4·83-s + 2·85-s − 18·89-s − 8·95-s − 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s + 1.66·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.312·41-s − 1.82·43-s − 0.274·53-s − 0.539·55-s − 0.520·59-s − 0.768·61-s − 0.744·65-s − 0.488·67-s − 0.949·71-s − 0.702·73-s − 1.80·79-s − 0.439·83-s + 0.216·85-s − 1.90·89-s − 0.820·95-s − 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−15.93349953614192, −15.72161853071825, −15.08688344582731, −14.30079966815919, −14.03581030474945, −13.30336757982223, −12.96519779038753, −11.95658181227936, −11.67564430955388, −11.28348091780818, −10.59974400232917, −9.857778044820740, −9.281076231532882, −8.725737852603378, −8.265277378676053, −7.386106250718670, −7.033018213534411, −6.126821549145555, −5.809692910709219, −4.899476276868807, −4.095510577977675, −3.568103806186419, −3.098638895928453, −1.672501557958054, −1.313702790044871, 0,
1.313702790044871, 1.672501557958054, 3.098638895928453, 3.568103806186419, 4.095510577977675, 4.899476276868807, 5.809692910709219, 6.126821549145555, 7.033018213534411, 7.386106250718670, 8.265277378676053, 8.725737852603378, 9.281076231532882, 9.857778044820740, 10.59974400232917, 11.28348091780818, 11.67564430955388, 11.95658181227936, 12.96519779038753, 13.30336757982223, 14.03581030474945, 14.30079966815919, 15.08688344582731, 15.72161853071825, 15.93349953614192
