| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s + 13-s + 16-s − 3·17-s + 7·19-s − 22-s + 4·23-s − 26-s − 2·29-s − 3·31-s − 32-s + 3·34-s + 2·37-s − 7·38-s + 6·41-s + 6·43-s + 44-s − 4·46-s + 4·47-s + 52-s − 2·53-s + 2·58-s + 4·59-s − 61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s + 0.277·13-s + 1/4·16-s − 0.727·17-s + 1.60·19-s − 0.213·22-s + 0.834·23-s − 0.196·26-s − 0.371·29-s − 0.538·31-s − 0.176·32-s + 0.514·34-s + 0.328·37-s − 1.13·38-s + 0.937·41-s + 0.914·43-s + 0.150·44-s − 0.589·46-s + 0.583·47-s + 0.138·52-s − 0.274·53-s + 0.262·58-s + 0.520·59-s − 0.128·61-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
| \(L(1)\) |
\(\approx\) |
\(2.056178784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.056178784\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−12.75561510631051, −12.18068799602130, −11.60221892066144, −11.35871151830996, −10.93920016967189, −10.36999130961227, −9.994007135280862, −9.297020135192769, −9.054822124852549, −8.829812954445586, −8.015426090812834, −7.512716343128617, −7.319809979607634, −6.745188100360315, −6.082566171714595, −5.801735504638409, −5.138574659140320, −4.620716725611845, −3.966154962758666, −3.438688686295564, −2.850352552324271, −2.358188240315104, −1.601560454187838, −1.069874046951209, −0.4865137407088737,
0.4865137407088737, 1.069874046951209, 1.601560454187838, 2.358188240315104, 2.850352552324271, 3.438688686295564, 3.966154962758666, 4.620716725611845, 5.138574659140320, 5.801735504638409, 6.082566171714595, 6.745188100360315, 7.319809979607634, 7.512716343128617, 8.015426090812834, 8.829812954445586, 9.054822124852549, 9.297020135192769, 9.994007135280862, 10.36999130961227, 10.93920016967189, 11.35871151830996, 11.60221892066144, 12.18068799602130, 12.75561510631051
