| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 12-s + 15-s + 16-s − 2·17-s + 18-s − 19-s + 20-s + 8·23-s + 24-s + 25-s + 27-s + 4·29-s + 30-s + 8·31-s + 32-s − 2·34-s + 36-s − 38-s + 40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 0.182·30-s + 1.43·31-s + 0.176·32-s − 0.342·34-s + 1/6·36-s − 0.162·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.596161136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.596161136\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| 107 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.08402040066795, −13.91832406812876, −13.23650864134251, −13.02066923916227, −12.33054597844197, −11.97109569092847, −11.21896045692869, −10.67968132986646, −10.42905206814412, −9.541272249997146, −9.220262448470551, −8.656512510288330, −8.022169472929062, −7.487694375360364, −6.873924538370299, −6.353114704705319, −5.935002938742862, −5.066791874569693, −4.627802324767137, −4.202022422526616, −3.206857519121482, −2.912589670867999, −2.278453272081115, −1.496985415651942, −0.7638129743886316,
0.7638129743886316, 1.496985415651942, 2.278453272081115, 2.912589670867999, 3.206857519121482, 4.202022422526616, 4.627802324767137, 5.066791874569693, 5.935002938742862, 6.353114704705319, 6.873924538370299, 7.487694375360364, 8.022169472929062, 8.656512510288330, 9.220262448470551, 9.541272249997146, 10.42905206814412, 10.67968132986646, 11.21896045692869, 11.97109569092847, 12.33054597844197, 13.02066923916227, 13.23650864134251, 13.91832406812876, 14.08402040066795
