| L(s) = 1 | + 2·2-s + 2·4-s + 5-s − 7-s + 2·10-s + 4·11-s − 2·13-s − 2·14-s − 4·16-s − 3·17-s − 8·19-s + 2·20-s + 8·22-s + 6·23-s − 4·25-s − 4·26-s − 2·28-s + 4·29-s + 6·31-s − 8·32-s − 6·34-s − 35-s − 3·37-s − 16·38-s − 41-s + 11·43-s + 8·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s + 0.632·10-s + 1.20·11-s − 0.554·13-s − 0.534·14-s − 16-s − 0.727·17-s − 1.83·19-s + 0.447·20-s + 1.70·22-s + 1.25·23-s − 4/5·25-s − 0.784·26-s − 0.377·28-s + 0.742·29-s + 1.07·31-s − 1.41·32-s − 1.02·34-s − 0.169·35-s − 0.493·37-s − 2.59·38-s − 0.156·41-s + 1.67·43-s + 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.264918073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.264918073\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−12.75190128426547053309293153894, −11.93218284814085692310865109390, −10.91981085800737069270016168181, −9.606127837407754601596009449174, −8.650153087392553143775802165667, −6.75587037635177377501803178294, −6.25471251304940803802164013919, −4.86590526026148497681960970818, −3.90517995805379232478125425669, −2.42236155159716987191201970387,
2.42236155159716987191201970387, 3.90517995805379232478125425669, 4.86590526026148497681960970818, 6.25471251304940803802164013919, 6.75587037635177377501803178294, 8.650153087392553143775802165667, 9.606127837407754601596009449174, 10.91981085800737069270016168181, 11.93218284814085692310865109390, 12.75190128426547053309293153894
