| L(s) = 1 | − 2-s + 2·3-s + 4-s + 2·5-s − 2·6-s + 7-s − 8-s + 9-s − 2·10-s + 4·11-s + 2·12-s − 2·13-s − 14-s + 4·15-s + 16-s − 4·17-s − 18-s + 2·19-s + 2·20-s + 2·21-s − 4·22-s − 2·24-s − 25-s + 2·26-s − 4·27-s + 28-s − 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.577·12-s − 0.554·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.458·19-s + 0.447·20-s + 0.436·21-s − 0.852·22-s − 0.408·24-s − 1/5·25-s + 0.392·26-s − 0.769·27-s + 0.188·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.696027844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.696027844\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 29 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−11.13201731544092088686923305461, −9.960610785746302660534895147946, −9.270052668138818769084969980892, −8.773935284643837596290484626400, −7.73714846452564120208314480356, −6.78660310245999746138955978208, −5.66418769884118381148197147273, −4.08563273524599149871051386292, −2.66451574357743769318009670602, −1.69144844888920009610292591649,
1.69144844888920009610292591649, 2.66451574357743769318009670602, 4.08563273524599149871051386292, 5.66418769884118381148197147273, 6.78660310245999746138955978208, 7.73714846452564120208314480356, 8.773935284643837596290484626400, 9.270052668138818769084969980892, 9.960610785746302660534895147946, 11.13201731544092088686923305461
