| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 7-s + 3·8-s + 9-s + 11-s + 2·12-s − 4·13-s − 14-s − 16-s − 4·17-s − 18-s − 2·21-s − 22-s + 4·23-s − 6·24-s + 4·26-s + 4·27-s − 28-s − 6·29-s + 10·31-s − 5·32-s − 2·33-s + 4·34-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.301·11-s + 0.577·12-s − 1.10·13-s − 0.267·14-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.436·21-s − 0.213·22-s + 0.834·23-s − 1.22·24-s + 0.784·26-s + 0.769·27-s − 0.188·28-s − 1.11·29-s + 1.79·31-s − 0.883·32-s − 0.348·33-s + 0.685·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.909929466225848181095380267427, −8.100012329111415913136438788578, −7.25001629514993403685065897796, −6.52215690743507933996587795029, −5.48176191593485566243882545926, −4.81403168222361608382158073717, −4.17701189100093347147152961961, −2.53981984920376994621368055964, −1.12960491671305260714837063392, 0,
1.12960491671305260714837063392, 2.53981984920376994621368055964, 4.17701189100093347147152961961, 4.81403168222361608382158073717, 5.48176191593485566243882545926, 6.52215690743507933996587795029, 7.25001629514993403685065897796, 8.100012329111415913136438788578, 8.909929466225848181095380267427
