| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 4·11-s − 2·13-s + 14-s + 16-s − 2·19-s + 20-s + 4·22-s − 2·23-s + 25-s − 2·26-s + 28-s + 6·29-s + 32-s + 35-s − 2·37-s − 2·38-s + 40-s + 6·41-s + 6·43-s + 4·44-s − 2·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.458·19-s + 0.223·20-s + 0.852·22-s − 0.417·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 0.176·32-s + 0.169·35-s − 0.328·37-s − 0.324·38-s + 0.158·40-s + 0.937·41-s + 0.914·43-s + 0.603·44-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.756144929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.756144929\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.10029357944010, −12.65029599667507, −12.23692755637291, −11.91310746229982, −11.21452568310358, −11.03438470316776, −10.29021046592587, −9.922187729024274, −9.406514009734880, −8.807388713371871, −8.469093986221945, −7.731832982659187, −7.276683922446389, −6.782041531346666, −6.184661945126364, −5.937425900984784, −5.290846992663631, −4.572750275879901, −4.388039885337117, −3.775252515671655, −3.038373717882296, −2.554890197602775, −1.870254645053073, −1.380474437791311, −0.6015695616475899,
0.6015695616475899, 1.380474437791311, 1.870254645053073, 2.554890197602775, 3.038373717882296, 3.775252515671655, 4.388039885337117, 4.572750275879901, 5.290846992663631, 5.937425900984784, 6.184661945126364, 6.782041531346666, 7.276683922446389, 7.731832982659187, 8.469093986221945, 8.807388713371871, 9.406514009734880, 9.922187729024274, 10.29021046592587, 11.03438470316776, 11.21452568310358, 11.91310746229982, 12.23692755637291, 12.65029599667507, 13.10029357944010
