| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 5·11-s + 12-s + 5·13-s − 15-s + 16-s + 4·17-s + 18-s + 7·19-s − 20-s − 5·22-s + 23-s + 24-s + 25-s + 5·26-s + 27-s − 30-s + 2·31-s + 32-s − 5·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s + 0.288·12-s + 1.38·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.60·19-s − 0.223·20-s − 1.06·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s + 0.192·27-s − 0.182·30-s + 0.359·31-s + 0.176·32-s − 0.870·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.140448834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.140448834\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−9.508323806371389828324953011682, −8.539621431094665013880367620876, −7.71924310223513706020182982975, −7.34861349178235057500600410969, −6.00648948433407704005183559854, −5.36455321315892538076371080248, −4.34679056765626736481568259216, −3.33743508603807834710909805134, −2.79055015422136948803015148393, −1.22656281915761565453295784614,
1.22656281915761565453295784614, 2.79055015422136948803015148393, 3.33743508603807834710909805134, 4.34679056765626736481568259216, 5.36455321315892538076371080248, 6.00648948433407704005183559854, 7.34861349178235057500600410969, 7.71924310223513706020182982975, 8.539621431094665013880367620876, 9.508323806371389828324953011682
