| L(s) = 1 | + 5-s − 4·7-s − 13-s + 17-s + 7·23-s + 25-s + 10·29-s + 3·31-s − 4·35-s + 12·37-s − 2·41-s − 10·43-s − 3·47-s + 9·49-s − 5·53-s + 14·59-s − 5·61-s − 65-s + 12·67-s + 8·71-s − 16·73-s − 3·79-s + 8·83-s + 85-s + 4·91-s + 12·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.277·13-s + 0.242·17-s + 1.45·23-s + 1/5·25-s + 1.85·29-s + 0.538·31-s − 0.676·35-s + 1.97·37-s − 0.312·41-s − 1.52·43-s − 0.437·47-s + 9/7·49-s − 0.686·53-s + 1.82·59-s − 0.640·61-s − 0.124·65-s + 1.46·67-s + 0.949·71-s − 1.87·73-s − 0.337·79-s + 0.878·83-s + 0.108·85-s + 0.419·91-s + 1.21·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−12.91688710977607, −12.78351725584907, −12.05578503925828, −11.61098379919517, −11.22865249711824, −10.45149980575801, −10.07904235110869, −9.896716051031689, −9.321066838680750, −8.919714485757090, −8.379048039652143, −7.887399023808187, −7.210227744602875, −6.761546438486809, −6.312645139176054, −6.171739038488241, −5.251853400620415, −4.962066417141892, −4.376710775033901, −3.621150720050937, −3.181000735954764, −2.686842234865623, −2.322709055164580, −1.239430251992093, −0.8492894634088224, 0,
0.8492894634088224, 1.239430251992093, 2.322709055164580, 2.686842234865623, 3.181000735954764, 3.621150720050937, 4.376710775033901, 4.962066417141892, 5.251853400620415, 6.171739038488241, 6.312645139176054, 6.761546438486809, 7.210227744602875, 7.887399023808187, 8.379048039652143, 8.919714485757090, 9.321066838680750, 9.896716051031689, 10.07904235110869, 10.45149980575801, 11.22865249711824, 11.61098379919517, 12.05578503925828, 12.78351725584907, 12.91688710977607
