| L(s) = 1 | + 2-s + 3·3-s + 4-s + 3·5-s + 3·6-s + 8-s + 6·9-s + 3·10-s − 2·11-s + 3·12-s + 5·13-s + 9·15-s + 16-s − 6·17-s + 6·18-s + 3·20-s − 2·22-s − 2·23-s + 3·24-s + 4·25-s + 5·26-s + 9·27-s + 6·29-s + 9·30-s + 10·31-s + 32-s − 6·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.34·5-s + 1.22·6-s + 0.353·8-s + 2·9-s + 0.948·10-s − 0.603·11-s + 0.866·12-s + 1.38·13-s + 2.32·15-s + 1/4·16-s − 1.45·17-s + 1.41·18-s + 0.670·20-s − 0.426·22-s − 0.417·23-s + 0.612·24-s + 4/5·25-s + 0.980·26-s + 1.73·27-s + 1.11·29-s + 1.64·30-s + 1.79·31-s + 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.585680399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.585680399\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−8.070627514425637178769125667667, −7.07716538698492996638658348794, −6.40634269350498332600322286103, −5.92498931196028374705526801624, −4.81762213986314591291099727820, −4.27825726597095235309150644259, −3.30258915419233219227704864492, −2.74539003178494066035862077804, −2.05403971801931958763833773416, −1.40530407403610158669497387516,
1.40530407403610158669497387516, 2.05403971801931958763833773416, 2.74539003178494066035862077804, 3.30258915419233219227704864492, 4.27825726597095235309150644259, 4.81762213986314591291099727820, 5.92498931196028374705526801624, 6.40634269350498332600322286103, 7.07716538698492996638658348794, 8.070627514425637178769125667667
