| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 6·13-s + 2·15-s + 2·19-s + 21-s − 25-s − 27-s − 8·29-s + 2·35-s − 2·37-s − 6·39-s − 2·41-s − 8·43-s − 2·45-s + 8·47-s + 49-s − 2·53-s − 2·57-s − 12·59-s + 4·61-s − 63-s − 12·65-s − 12·67-s − 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s + 0.458·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s + 0.338·35-s − 0.328·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 0.264·57-s − 1.56·59-s + 0.512·61-s − 0.125·63-s − 1.48·65-s − 1.46·67-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5891827456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5891827456\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.58894077752464, −13.30741604457639, −12.85705739097747, −12.11926557958847, −11.83415838114041, −11.39877669335491, −10.80236378169213, −10.60758456607021, −9.844601230902770, −9.304413461919809, −8.770688712788971, −8.321712662748036, −7.526924860149732, −7.451080086310695, −6.532056164233998, −6.250791697010591, −5.564060100850830, −5.191894342539365, −4.232073474519103, −3.961817514090528, −3.400647872719444, −2.832459253827939, −1.703473737852224, −1.269900638651154, −0.2660512846131322,
0.2660512846131322, 1.269900638651154, 1.703473737852224, 2.832459253827939, 3.400647872719444, 3.961817514090528, 4.232073474519103, 5.191894342539365, 5.564060100850830, 6.250791697010591, 6.532056164233998, 7.451080086310695, 7.526924860149732, 8.321712662748036, 8.770688712788971, 9.304413461919809, 9.844601230902770, 10.60758456607021, 10.80236378169213, 11.39877669335491, 11.83415838114041, 12.11926557958847, 12.85705739097747, 13.30741604457639, 13.58894077752464
