| L(s) = 1 | − 2-s + 4-s + 3·5-s − 4·7-s − 8-s − 3·10-s + 3·11-s + 5·13-s + 4·14-s + 16-s + 3·17-s − 7·19-s + 3·20-s − 3·22-s + 6·23-s + 4·25-s − 5·26-s − 4·28-s + 6·29-s + 31-s − 32-s − 3·34-s − 12·35-s + 2·37-s + 7·38-s − 3·40-s − 12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.51·7-s − 0.353·8-s − 0.948·10-s + 0.904·11-s + 1.38·13-s + 1.06·14-s + 1/4·16-s + 0.727·17-s − 1.60·19-s + 0.670·20-s − 0.639·22-s + 1.25·23-s + 4/5·25-s − 0.980·26-s − 0.755·28-s + 1.11·29-s + 0.179·31-s − 0.176·32-s − 0.514·34-s − 2.02·35-s + 0.328·37-s + 1.13·38-s − 0.474·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.256064199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256064199\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−10.34853733995905187730822977685, −9.964335668702123063898779925679, −9.008385650207468477029609714011, −8.560114382003000320260887825597, −6.77081160640587022382984853685, −6.46383969687873120831147421873, −5.63088115150215301432006425784, −3.81465163434429469092349648264, −2.65868474069055778441649444029, −1.21274221758651936189138016017,
1.21274221758651936189138016017, 2.65868474069055778441649444029, 3.81465163434429469092349648264, 5.63088115150215301432006425784, 6.46383969687873120831147421873, 6.77081160640587022382984853685, 8.560114382003000320260887825597, 9.008385650207468477029609714011, 9.964335668702123063898779925679, 10.34853733995905187730822977685
