| L(s) = 1 | + 2-s + 4-s + 4·5-s + 8-s + 4·10-s + 11-s + 13-s + 16-s + 4·17-s + 4·20-s + 22-s + 11·25-s + 26-s + 10·31-s + 32-s + 4·34-s + 10·37-s + 4·40-s + 2·41-s + 6·43-s + 44-s − 12·47-s + 11·50-s + 52-s + 2·53-s + 4·55-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.353·8-s + 1.26·10-s + 0.301·11-s + 0.277·13-s + 1/4·16-s + 0.970·17-s + 0.894·20-s + 0.213·22-s + 11/5·25-s + 0.196·26-s + 1.79·31-s + 0.176·32-s + 0.685·34-s + 1.64·37-s + 0.632·40-s + 0.312·41-s + 0.914·43-s + 0.150·44-s − 1.75·47-s + 1.55·50-s + 0.138·52-s + 0.274·53-s + 0.539·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.521598972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.521598972\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.62288265924177, −13.08551961710473, −12.72835152638550, −12.19322065469088, −11.66782324719779, −11.08171076702284, −10.65414683784305, −9.946507234128958, −9.795057071602037, −9.344074599464166, −8.630167267296647, −8.084191724858299, −7.555944778083551, −6.776167568398018, −6.268064261531276, −6.134320707062679, −5.496825410752016, −4.959092700965399, −4.558876983280835, −3.707821971535853, −3.179808098929712, −2.407574184539418, −2.212683095566424, −1.183247085758524, −0.9568425461443429,
0.9568425461443429, 1.183247085758524, 2.212683095566424, 2.407574184539418, 3.179808098929712, 3.707821971535853, 4.558876983280835, 4.959092700965399, 5.496825410752016, 6.134320707062679, 6.268064261531276, 6.776167568398018, 7.555944778083551, 8.084191724858299, 8.630167267296647, 9.344074599464166, 9.795057071602037, 9.946507234128958, 10.65414683784305, 11.08171076702284, 11.66782324719779, 12.19322065469088, 12.72835152638550, 13.08551961710473, 13.62288265924177
