| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 4·13-s + 14-s + 16-s − 4·19-s − 20-s + 25-s + 4·26-s − 28-s − 3·29-s − 5·31-s − 32-s + 35-s + 7·37-s + 4·38-s + 40-s + 6·41-s − 10·43-s + 3·47-s + 49-s − 50-s − 4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 0.557·29-s − 0.898·31-s − 0.176·32-s + 0.169·35-s + 1.15·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s − 1.52·43-s + 0.437·47-s + 1/7·49-s − 0.141·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−13.10368736411761, −12.84565070434384, −12.49233779370553, −11.90429647258731, −11.41067585265659, −11.03160765779694, −10.53956414530585, −9.981839493299117, −9.600516232279976, −9.133534876108711, −8.706908918736169, −7.958587122675764, −7.779480383617202, −7.222681171940249, −6.633220837477470, −6.329543276384820, −5.597252472016017, −5.050973736294026, −4.492927645255037, −3.822547572832585, −3.383778821192051, −2.547240371141798, −2.250980151011599, −1.471690635124756, −0.5914341591415261, 0,
0.5914341591415261, 1.471690635124756, 2.250980151011599, 2.547240371141798, 3.383778821192051, 3.822547572832585, 4.492927645255037, 5.050973736294026, 5.597252472016017, 6.329543276384820, 6.633220837477470, 7.222681171940249, 7.779480383617202, 7.958587122675764, 8.706908918736169, 9.133534876108711, 9.600516232279976, 9.981839493299117, 10.53956414530585, 11.03160765779694, 11.41067585265659, 11.90429647258731, 12.49233779370553, 12.84565070434384, 13.10368736411761
