| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s − 13-s + 16-s + 2·17-s − 2·19-s − 20-s + 22-s + 2·23-s − 4·25-s + 26-s + 7·29-s + 3·31-s − 32-s − 2·34-s − 2·37-s + 2·38-s + 40-s − 8·41-s − 8·43-s − 44-s − 2·46-s + 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.458·19-s − 0.223·20-s + 0.213·22-s + 0.417·23-s − 4/5·25-s + 0.196·26-s + 1.29·29-s + 0.538·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s + 0.324·38-s + 0.158·40-s − 1.24·41-s − 1.21·43-s − 0.150·44-s − 0.294·46-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9706982781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9706982781\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−16.58586938668035, −15.81096457645879, −15.44232547123464, −14.95313690245690, −14.21288642815555, −13.61584865487483, −12.96218944345176, −12.18090231948082, −11.86821483861089, −11.24768020975552, −10.43547245000408, −10.14578234223283, −9.457841208049120, −8.708370505884880, −8.120412834476169, −7.790859682544867, −6.803064751634443, −6.560819137739500, −5.500468483194962, −4.928539306764707, −4.020492447098015, −3.244839760646103, −2.485773796619173, −1.563779429901522, −0.5102877445526610,
0.5102877445526610, 1.563779429901522, 2.485773796619173, 3.244839760646103, 4.020492447098015, 4.928539306764707, 5.500468483194962, 6.560819137739500, 6.803064751634443, 7.790859682544867, 8.120412834476169, 8.708370505884880, 9.457841208049120, 10.14578234223283, 10.43547245000408, 11.24768020975552, 11.86821483861089, 12.18090231948082, 12.96218944345176, 13.61584865487483, 14.21288642815555, 14.95313690245690, 15.44232547123464, 15.81096457645879, 16.58586938668035
