| L(s) = 1 | − 2-s + 4-s − 8-s + 2·11-s + 13-s + 16-s − 5·17-s − 4·19-s − 2·22-s − 3·23-s − 26-s + 8·29-s − 31-s − 32-s + 5·34-s + 6·37-s + 4·38-s − 3·41-s + 10·43-s + 2·44-s + 3·46-s − 3·47-s + 52-s − 8·58-s − 2·61-s + 62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.603·11-s + 0.277·13-s + 1/4·16-s − 1.21·17-s − 0.917·19-s − 0.426·22-s − 0.625·23-s − 0.196·26-s + 1.48·29-s − 0.179·31-s − 0.176·32-s + 0.857·34-s + 0.986·37-s + 0.648·38-s − 0.468·41-s + 1.52·43-s + 0.301·44-s + 0.442·46-s − 0.437·47-s + 0.138·52-s − 1.05·58-s − 0.256·61-s + 0.127·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.713835863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713835863\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−12.62740209640697, −12.26357997170410, −11.71012912490091, −11.18464674100838, −10.97792466823721, −10.41539385378563, −9.916775651460498, −9.525071658119185, −8.894998189285026, −8.684911914769228, −8.147570128752052, −7.748506827047646, −7.026734324958598, −6.668589935251454, −6.215293242547741, −5.899556639469472, −5.055119556323281, −4.482628272471372, −4.104535957651687, −3.507369536773015, −2.761856968864695, −2.255795780465861, −1.797787742696117, −0.9859375615715374, −0.4519340955070770,
0.4519340955070770, 0.9859375615715374, 1.797787742696117, 2.255795780465861, 2.761856968864695, 3.507369536773015, 4.104535957651687, 4.482628272471372, 5.055119556323281, 5.899556639469472, 6.215293242547741, 6.668589935251454, 7.026734324958598, 7.748506827047646, 8.147570128752052, 8.684911914769228, 8.894998189285026, 9.525071658119185, 9.916775651460498, 10.41539385378563, 10.97792466823721, 11.18464674100838, 11.71012912490091, 12.26357997170410, 12.62740209640697
