| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s + 10-s − 13-s − 16-s + 2·17-s − 4·19-s + 20-s − 8·23-s + 25-s + 26-s − 6·29-s − 5·32-s − 2·34-s + 2·37-s + 4·38-s − 3·40-s + 6·41-s − 8·43-s + 8·46-s + 4·47-s − 50-s + 52-s + 6·53-s + 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s − 0.277·13-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s + 0.196·26-s − 1.11·29-s − 0.883·32-s − 0.342·34-s + 0.328·37-s + 0.648·38-s − 0.474·40-s + 0.937·41-s − 1.21·43-s + 1.17·46-s + 0.583·47-s − 0.141·50-s + 0.138·52-s + 0.824·53-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.53721648160099, −14.90565709720113, −14.36434047228873, −13.99933861103712, −13.25266610518493, −12.81193760952533, −12.29435677204030, −11.63268749462684, −11.10240974304765, −10.47598652932811, −9.954209381250885, −9.574708838191068, −8.865983656676245, −8.345712573600049, −7.896809898708960, −7.412776874189253, −6.727596926423207, −5.943116547793296, −5.342423988942120, −4.622290617627256, −3.956887431552243, −3.621028945780341, −2.415312399625018, −1.807798627732522, −0.7795237758628981, 0,
0.7795237758628981, 1.807798627732522, 2.415312399625018, 3.621028945780341, 3.956887431552243, 4.622290617627256, 5.342423988942120, 5.943116547793296, 6.727596926423207, 7.412776874189253, 7.896809898708960, 8.345712573600049, 8.865983656676245, 9.574708838191068, 9.954209381250885, 10.47598652932811, 11.10240974304765, 11.63268749462684, 12.29435677204030, 12.81193760952533, 13.25266610518493, 13.99933861103712, 14.36434047228873, 14.90565709720113, 15.53721648160099
