| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s + 7-s − 8-s + 9-s + 2·12-s − 14-s + 16-s − 18-s − 2·19-s + 2·21-s + 6·23-s − 2·24-s − 4·27-s + 28-s + 6·29-s − 8·31-s − 32-s + 36-s − 10·37-s + 2·38-s + 12·41-s − 2·42-s + 4·43-s − 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.577·12-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.436·21-s + 1.25·23-s − 0.408·24-s − 0.769·27-s + 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 1/6·36-s − 1.64·37-s + 0.324·38-s + 1.87·41-s − 0.308·42-s + 0.609·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.582783921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582783921\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.35478976032820, −14.09907584180873, −13.19831782851946, −13.01942876843409, −12.25480212718663, −11.76917021977286, −11.13254076532948, −10.59697395734457, −10.32302006738435, −9.391031578131678, −9.103063419975848, −8.727801236801871, −8.231621851490126, −7.647247041072118, −7.187755949172109, −6.732774629006391, −5.816999975730042, −5.409626217500099, −4.521133818386372, −3.949111005493545, −3.199431239964012, −2.723452518049712, −2.091319789068753, −1.452725468900023, −0.5709326933053010,
0.5709326933053010, 1.452725468900023, 2.091319789068753, 2.723452518049712, 3.199431239964012, 3.949111005493545, 4.521133818386372, 5.409626217500099, 5.816999975730042, 6.732774629006391, 7.187755949172109, 7.647247041072118, 8.231621851490126, 8.727801236801871, 9.103063419975848, 9.391031578131678, 10.32302006738435, 10.59697395734457, 11.13254076532948, 11.76917021977286, 12.25480212718663, 13.01942876843409, 13.19831782851946, 14.09907584180873, 14.35478976032820
