| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s + 10-s − 2·13-s − 16-s − 2·17-s − 4·19-s + 20-s + 25-s + 2·26-s + 6·29-s − 4·31-s − 5·32-s + 2·34-s + 6·37-s + 4·38-s − 3·40-s + 10·41-s − 12·43-s − 7·49-s − 50-s + 2·52-s − 6·53-s − 6·58-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s − 0.554·13-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.986·37-s + 0.648·38-s − 0.474·40-s + 1.56·41-s − 1.82·43-s − 49-s − 0.141·50-s + 0.277·52-s − 0.824·53-s − 0.787·58-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.65026204030446, −12.53184005272181, −11.64910983327853, −11.27047362603373, −10.97642250780402, −10.33059425770532, −10.00227914212534, −9.517610746706932, −9.159740282862802, −8.508092132665494, −8.296678215221551, −7.855837517423836, −7.342358135252009, −6.807124888182586, −6.420266561554702, −5.783839432241840, −5.045734399096664, −4.772993829693532, −4.255651323767848, −3.817257069002448, −3.165734201614297, −2.479025208804856, −1.963114403618115, −1.255237822730660, −0.5794502746887954, 0,
0.5794502746887954, 1.255237822730660, 1.963114403618115, 2.479025208804856, 3.165734201614297, 3.817257069002448, 4.255651323767848, 4.772993829693532, 5.045734399096664, 5.783839432241840, 6.420266561554702, 6.807124888182586, 7.342358135252009, 7.855837517423836, 8.296678215221551, 8.508092132665494, 9.159740282862802, 9.517610746706932, 10.00227914212534, 10.33059425770532, 10.97642250780402, 11.27047362603373, 11.64910983327853, 12.53184005272181, 12.65026204030446
