| L(s) = 1 | − 5-s − 7-s + 6·11-s − 4·13-s + 2·19-s + 25-s + 6·29-s + 10·31-s + 35-s − 2·37-s − 6·41-s − 4·43-s + 49-s + 12·53-s − 6·55-s − 14·61-s + 4·65-s − 4·67-s + 6·71-s + 4·73-s − 6·77-s + 16·79-s + 12·83-s − 6·89-s + 4·91-s − 2·95-s + 16·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.80·11-s − 1.10·13-s + 0.458·19-s + 1/5·25-s + 1.11·29-s + 1.79·31-s + 0.169·35-s − 0.328·37-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 1.64·53-s − 0.809·55-s − 1.79·61-s + 0.496·65-s − 0.488·67-s + 0.712·71-s + 0.468·73-s − 0.683·77-s + 1.80·79-s + 1.31·83-s − 0.635·89-s + 0.419·91-s − 0.205·95-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−12.34082599485676, −12.24542107831570, −11.91477825574256, −11.66116584079267, −10.91251285837669, −10.44679381277804, −9.965318820777505, −9.545772019619672, −9.209663408547628, −8.594744885496540, −8.254743598976432, −7.687971148652362, −7.081031759928277, −6.762570911083295, −6.383310342823356, −5.894648793617450, −5.067022866714917, −4.738687047693320, −4.278307817710365, −3.615628682538095, −3.302254394186377, −2.624565140771285, −2.083230463500185, −1.225641137075653, −0.8577754763724076, 0,
0.8577754763724076, 1.225641137075653, 2.083230463500185, 2.624565140771285, 3.302254394186377, 3.615628682538095, 4.278307817710365, 4.738687047693320, 5.067022866714917, 5.894648793617450, 6.383310342823356, 6.762570911083295, 7.081031759928277, 7.687971148652362, 8.254743598976432, 8.594744885496540, 9.209663408547628, 9.545772019619672, 9.965318820777505, 10.44679381277804, 10.91251285837669, 11.66116584079267, 11.91477825574256, 12.24542107831570, 12.34082599485676
