| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 2·11-s + 4·13-s + 14-s + 16-s − 4·19-s − 20-s + 2·22-s − 4·23-s + 25-s − 4·26-s − 28-s − 8·29-s + 4·31-s − 32-s + 35-s − 2·37-s + 4·38-s + 40-s − 2·41-s − 4·43-s − 2·44-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 − 0.603·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 0.426·22-s − 0.834·23-s + 1/5·25-s − 0.784·26-s − 0.188·28-s − 1.48·29-s + 0.718·31-s − 0.176·32-s + 0.169·35-s − 0.328·37-s + 0.648·38-s + 0.158·40-s − 0.312·41-s − 0.609·43-s − 0.301·44-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.42136439488862, −13.15218821853610, −12.68059616239453, −12.18149735947471, −11.55631027791445, −11.22501222585580, −10.82761122043269, −10.20482408804049, −9.947252300849898, −9.341977243326769, −8.759572575222008, −8.275294988956571, −8.146051141593351, −7.372468080410637, −6.992966126864762, −6.361271978412438, −5.965508855631848, −5.475913220161521, −4.695463705006664, −4.128996836223969, −3.562846040269040, −3.101279097904858, −2.383570680814751, −1.744639914312255, −1.173598213324121, 0, 0,
1.173598213324121, 1.744639914312255, 2.383570680814751, 3.101279097904858, 3.562846040269040, 4.128996836223969, 4.695463705006664, 5.475913220161521, 5.965508855631848, 6.361271978412438, 6.992966126864762, 7.372468080410637, 8.146051141593351, 8.275294988956571, 8.759572575222008, 9.341977243326769, 9.947252300849898, 10.20482408804049, 10.82761122043269, 11.22501222585580, 11.55631027791445, 12.18149735947471, 12.68059616239453, 13.15218821853610, 13.42136439488862
