| L(s) = 1 | − 3-s + 9-s + 3·13-s + 2·17-s − 19-s + 2·23-s − 27-s + 8·29-s − 8·31-s − 7·37-s − 3·39-s + 8·43-s − 10·47-s − 2·51-s + 14·53-s + 57-s − 10·59-s − 7·61-s + 5·67-s − 2·69-s − 12·71-s − 11·73-s − 7·79-s + 81-s − 14·83-s − 8·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.832·13-s + 0.485·17-s − 0.229·19-s + 0.417·23-s − 0.192·27-s + 1.48·29-s − 1.43·31-s − 1.15·37-s − 0.480·39-s + 1.21·43-s − 1.45·47-s − 0.280·51-s + 1.92·53-s + 0.132·57-s − 1.30·59-s − 0.896·61-s + 0.610·67-s − 0.240·69-s − 1.42·71-s − 1.28·73-s − 0.787·79-s + 1/9·81-s − 1.53·83-s − 0.857·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.374632077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374632077\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−12.84930652130378, −12.48471799937566, −11.88358518728922, −11.60222913097469, −11.02449428088111, −10.56236054865364, −10.29552204646474, −9.748639646728831, −9.056896107638494, −8.733896329124365, −8.316202918264227, −7.546191697430215, −7.246240820750751, −6.702815966621516, −6.112438095184554, −5.781998914743921, −5.246691967710354, −4.676867479936391, −4.186544606524896, −3.584173723227354, −3.068253465388014, −2.444624830515103, −1.517364185778074, −1.273695715504261, −0.3489269157994069,
0.3489269157994069, 1.273695715504261, 1.517364185778074, 2.444624830515103, 3.068253465388014, 3.584173723227354, 4.186544606524896, 4.676867479936391, 5.246691967710354, 5.781998914743921, 6.112438095184554, 6.702815966621516, 7.246240820750751, 7.546191697430215, 8.316202918264227, 8.733896329124365, 9.056896107638494, 9.748639646728831, 10.29552204646474, 10.56236054865364, 11.02449428088111, 11.60222913097469, 11.88358518728922, 12.48471799937566, 12.84930652130378
