| L(s) = 1 | + 2-s + 4-s + 3·5-s − 2·7-s + 8-s + 3·10-s + 6·11-s − 13-s − 2·14-s + 16-s + 6·17-s − 2·19-s + 3·20-s + 6·22-s + 4·25-s − 26-s − 2·28-s + 9·29-s − 4·31-s + 32-s + 6·34-s − 6·35-s − 2·37-s − 2·38-s + 3·40-s + 9·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s + 0.353·8-s + 0.948·10-s + 1.80·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.670·20-s + 1.27·22-s + 4/5·25-s − 0.196·26-s − 0.377·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 1.01·35-s − 0.328·37-s − 0.324·38-s + 0.474·40-s + 1.40·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.055495664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.055495664\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.40359356540113353963997146172, −6.76209924349764538841245111250, −6.09245442774062358850687959011, −5.93027148797431811844322458206, −4.97301092567470492640403963987, −4.19469456868339768946759192820, −3.41037328204235527468971451644, −2.74197465625021747022776144700, −1.77020260442271018297215635960, −1.04577926095690096882809558016,
1.04577926095690096882809558016, 1.77020260442271018297215635960, 2.74197465625021747022776144700, 3.41037328204235527468971451644, 4.19469456868339768946759192820, 4.97301092567470492640403963987, 5.93027148797431811844322458206, 6.09245442774062358850687959011, 6.76209924349764538841245111250, 7.40359356540113353963997146172
