| L(s) = 1 | − 2-s + 4-s − 8-s + 4·11-s + 13-s + 16-s − 6·17-s + 4·19-s − 4·22-s − 26-s + 4·29-s − 4·31-s − 32-s + 6·34-s − 12·37-s − 4·38-s + 12·41-s + 8·43-s + 4·44-s − 2·47-s + 52-s + 8·53-s − 4·58-s − 4·59-s − 10·61-s + 4·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.20·11-s + 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.852·22-s − 0.196·26-s + 0.742·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s − 1.97·37-s − 0.648·38-s + 1.87·41-s + 1.21·43-s + 0.603·44-s − 0.291·47-s + 0.138·52-s + 1.09·53-s − 0.525·58-s − 0.520·59-s − 1.28·61-s + 0.508·62-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
| \(L(1)\) |
\(\approx\) |
\(2.159656701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159656701\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.56669169143283, −12.24443058950485, −11.66746366503944, −11.34459217914997, −10.86265444755058, −10.47470918370318, −9.954236887049017, −9.245976041953955, −9.048209661836947, −8.834136768943548, −8.123682638754797, −7.565415310223235, −7.143697778350395, −6.711964321634330, −6.206230004988434, −5.821748624222181, −5.097239014051508, −4.564364495233499, −3.953977457491548, −3.534600348107713, −2.876322777044548, −2.189374633249379, −1.745198554759968, −1.012898643758162, −0.5085215890882216,
0.5085215890882216, 1.012898643758162, 1.745198554759968, 2.189374633249379, 2.876322777044548, 3.534600348107713, 3.953977457491548, 4.564364495233499, 5.097239014051508, 5.821748624222181, 6.206230004988434, 6.711964321634330, 7.143697778350395, 7.565415310223235, 8.123682638754797, 8.834136768943548, 9.048209661836947, 9.245976041953955, 9.954236887049017, 10.47470918370318, 10.86265444755058, 11.34459217914997, 11.66746366503944, 12.24443058950485, 12.56669169143283
