| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s + 2·13-s + 14-s + 16-s − 4·17-s + 4·19-s − 20-s − 22-s − 4·23-s − 4·25-s − 2·26-s − 28-s + 2·29-s − 5·31-s − 32-s + 4·34-s + 35-s + 8·37-s − 4·38-s + 40-s + 41-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 + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.917·19-s − 0.223·20-s − 0.213·22-s − 0.834·23-s − 4/5·25-s − 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.898·31-s − 0.176·32-s + 0.685·34-s + 0.169·35-s + 1.31·37-s − 0.648·38-s + 0.158·40-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−8.700019904160032547114765269254, −7.896315211732471771349163315903, −7.30964817955397906768521459416, −6.36220100362852171034641416074, −5.78676090235186788843455685481, −4.46969343217657062525490222266, −3.66331656518873575452699806263, −2.63412998189801327801840670109, −1.41687392997631507529266225221, 0,
1.41687392997631507529266225221, 2.63412998189801327801840670109, 3.66331656518873575452699806263, 4.46969343217657062525490222266, 5.78676090235186788843455685481, 6.36220100362852171034641416074, 7.30964817955397906768521459416, 7.896315211732471771349163315903, 8.700019904160032547114765269254
