| L(s) = 1 | + 3-s − 3·5-s + 9-s − 2·11-s − 3·15-s − 7·19-s + 7·23-s + 4·25-s + 27-s + 3·29-s + 5·31-s − 2·33-s − 4·37-s − 6·41-s + 11·43-s − 3·45-s + 3·47-s − 9·53-s + 6·55-s − 7·57-s + 8·59-s + 14·61-s + 2·67-s + 7·69-s + 3·73-s + 4·75-s + 3·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.603·11-s − 0.774·15-s − 1.60·19-s + 1.45·23-s + 4/5·25-s + 0.192·27-s + 0.557·29-s + 0.898·31-s − 0.348·33-s − 0.657·37-s − 0.937·41-s + 1.67·43-s − 0.447·45-s + 0.437·47-s − 1.23·53-s + 0.809·55-s − 0.927·57-s + 1.04·59-s + 1.79·61-s + 0.244·67-s + 0.842·69-s + 0.351·73-s + 0.461·75-s + 0.337·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.63406101995216, −12.28174083893121, −11.83353146376992, −11.12469714354498, −11.01699391154435, −10.47158062238873, −10.02171492785258, −9.446510696224049, −8.857969697680182, −8.469913195956953, −8.203250205509530, −7.777328950645366, −7.181827225642455, −6.789777393725455, −6.473025304801697, −5.576150703560380, −5.169093714939579, −4.441317879248690, −4.293727858836946, −3.670673013124776, −3.171205972611597, −2.606450093206192, −2.228058851049358, −1.319248198226303, −0.6827583705843432, 0,
0.6827583705843432, 1.319248198226303, 2.228058851049358, 2.606450093206192, 3.171205972611597, 3.670673013124776, 4.293727858836946, 4.441317879248690, 5.169093714939579, 5.576150703560380, 6.473025304801697, 6.789777393725455, 7.181827225642455, 7.777328950645366, 8.203250205509530, 8.469913195956953, 8.857969697680182, 9.446510696224049, 10.02171492785258, 10.47158062238873, 11.01699391154435, 11.12469714354498, 11.83353146376992, 12.28174083893121, 12.63406101995216
