| L(s) = 1 | − 2-s + 4-s − 5-s − 5·7-s − 8-s + 10-s − 4·11-s − 2·13-s + 5·14-s + 16-s + 2·17-s + 8·19-s − 20-s + 4·22-s − 3·23-s − 4·25-s + 2·26-s − 5·28-s + 4·31-s − 32-s − 2·34-s + 5·35-s − 4·37-s − 8·38-s + 40-s + 2·41-s − 6·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.88·7-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 0.554·13-s + 1.33·14-s + 1/4·16-s + 0.485·17-s + 1.83·19-s − 0.223·20-s + 0.852·22-s − 0.625·23-s − 4/5·25-s + 0.392·26-s − 0.944·28-s + 0.718·31-s − 0.176·32-s − 0.342·34-s + 0.845·35-s − 0.657·37-s − 1.29·38-s + 0.158·40-s + 0.312·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7526075857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7526075857\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 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
−13.34435517688611, −12.94605854639994, −12.36656539313752, −11.93311830592679, −11.71112587827997, −10.91004016762624, −10.20040560137866, −10.07905075385176, −9.695045413631203, −9.214596726687447, −8.576954833185403, −7.952469697302995, −7.603216660706601, −7.103527570211942, −6.704285652464676, −5.997695478158423, −5.473905745147422, −5.167047007154675, −4.075302067084081, −3.635194850531137, −2.995857776848015, −2.699256668481002, −1.968756585594896, −0.8448227567334723, −0.3820900125465381,
0.3820900125465381, 0.8448227567334723, 1.968756585594896, 2.699256668481002, 2.995857776848015, 3.635194850531137, 4.075302067084081, 5.167047007154675, 5.473905745147422, 5.997695478158423, 6.704285652464676, 7.103527570211942, 7.603216660706601, 7.952469697302995, 8.576954833185403, 9.214596726687447, 9.695045413631203, 10.07905075385176, 10.20040560137866, 10.91004016762624, 11.71112587827997, 11.93311830592679, 12.36656539313752, 12.94605854639994, 13.34435517688611
