| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s − 11-s − 13-s − 14-s − 16-s + 5·19-s + 22-s + 26-s − 28-s + 5·29-s − 2·31-s − 5·32-s − 4·37-s − 5·38-s + 5·41-s − 9·43-s + 44-s − 6·47-s − 6·49-s + 52-s − 2·53-s + 3·56-s − 5·58-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 0.301·11-s − 0.277·13-s − 0.267·14-s − 1/4·16-s + 1.14·19-s + 0.213·22-s + 0.196·26-s − 0.188·28-s + 0.928·29-s − 0.359·31-s − 0.883·32-s − 0.657·37-s − 0.811·38-s + 0.780·41-s − 1.37·43-s + 0.150·44-s − 0.875·47-s − 6/7·49-s + 0.138·52-s − 0.274·53-s + 0.400·56-s − 0.656·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−13.87426270856446, −13.32500764611219, −12.93584603969461, −12.31180970664388, −11.88891038766522, −11.24556221661495, −10.86631953114556, −10.31382297349658, −9.751344522328023, −9.540394660190053, −8.949510850351012, −8.322955197085984, −7.969657714797587, −7.661767731262887, −6.812200432255259, −6.611142300311935, −5.539768895004713, −5.204772753962651, −4.781964305721787, −4.139110012837771, −3.466737396792439, −2.917421856308301, −2.048543698867785, −1.456022084149149, −0.7907398212175567, 0,
0.7907398212175567, 1.456022084149149, 2.048543698867785, 2.917421856308301, 3.466737396792439, 4.139110012837771, 4.781964305721787, 5.204772753962651, 5.539768895004713, 6.611142300311935, 6.812200432255259, 7.661767731262887, 7.969657714797587, 8.322955197085984, 8.949510850351012, 9.540394660190053, 9.751344522328023, 10.31382297349658, 10.86631953114556, 11.24556221661495, 11.88891038766522, 12.31180970664388, 12.93584603969461, 13.32500764611219, 13.87426270856446
