| L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s + 7-s + 8-s − 2·9-s − 4·10-s + 12-s − 2·13-s + 14-s − 4·15-s + 16-s − 2·18-s + 3·19-s − 4·20-s + 21-s − 4·23-s + 24-s + 11·25-s − 2·26-s − 5·27-s + 28-s − 29-s − 4·30-s + 9·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 1.26·10-s + 0.288·12-s − 0.554·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s − 0.471·18-s + 0.688·19-s − 0.894·20-s + 0.218·21-s − 0.834·23-s + 0.204·24-s + 11/5·25-s − 0.392·26-s − 0.962·27-s + 0.188·28-s − 0.185·29-s − 0.730·30-s + 1.61·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.362771144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.362771144\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−8.267924804472415122122038555599, −7.63768846359051885320377750328, −7.28436575889116455955622799622, −6.17114861868817108814561157520, −5.30923996702738821790360609592, −4.38211018833650985019627393787, −3.96288670793729893993914350994, −3.04296579586378534444595425856, −2.42282315290469378664282284912, −0.75204211022204226740740650201,
0.75204211022204226740740650201, 2.42282315290469378664282284912, 3.04296579586378534444595425856, 3.96288670793729893993914350994, 4.38211018833650985019627393787, 5.30923996702738821790360609592, 6.17114861868817108814561157520, 7.28436575889116455955622799622, 7.63768846359051885320377750328, 8.267924804472415122122038555599
