| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s − 11-s − 2·13-s + 16-s + 3·17-s − 7·19-s + 2·20-s − 22-s − 7·23-s − 25-s − 2·26-s − 5·29-s − 2·31-s + 32-s + 3·34-s + 3·37-s − 7·38-s + 2·40-s − 6·41-s + 11·43-s − 44-s − 7·46-s + 7·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.727·17-s − 1.60·19-s + 0.447·20-s − 0.213·22-s − 1.45·23-s − 1/5·25-s − 0.392·26-s − 0.928·29-s − 0.359·31-s + 0.176·32-s + 0.514·34-s + 0.493·37-s − 1.13·38-s + 0.316·40-s − 0.937·41-s + 1.67·43-s − 0.150·44-s − 1.03·46-s + 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−7.37728934655450291128133177848, −6.33963802465275997783800622425, −5.93377691761446650375864363448, −5.44064540083757532260619783697, −4.48406013343064826193939510736, −3.98892283662584882145313796126, −2.95763204442435622466295731178, −2.19796085245218112018013778894, −1.62405555653882317648926214414, 0,
1.62405555653882317648926214414, 2.19796085245218112018013778894, 2.95763204442435622466295731178, 3.98892283662584882145313796126, 4.48406013343064826193939510736, 5.44064540083757532260619783697, 5.93377691761446650375864363448, 6.33963802465275997783800622425, 7.37728934655450291128133177848
