| L(s) = 1 | + 2·3-s − 5-s − 2·7-s + 9-s − 2·15-s + 3·17-s − 6·19-s − 4·21-s + 4·23-s − 4·25-s − 4·27-s − 29-s − 4·31-s + 2·35-s − 9·37-s − 41-s − 4·43-s − 45-s + 6·47-s − 3·49-s + 6·51-s − 9·53-s − 12·57-s + 6·59-s − 5·61-s − 2·63-s − 6·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.516·15-s + 0.727·17-s − 1.37·19-s − 0.872·21-s + 0.834·23-s − 4/5·25-s − 0.769·27-s − 0.185·29-s − 0.718·31-s + 0.338·35-s − 1.47·37-s − 0.156·41-s − 0.609·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.840·51-s − 1.23·53-s − 1.58·57-s + 0.781·59-s − 0.640·61-s − 0.251·63-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−12.95802206387801, −12.73524292217642, −12.33346347133028, −11.66187595207706, −11.30218552940822, −10.72398929553510, −10.21623067314784, −9.856576056358523, −9.256862598057485, −8.960582647131557, −8.433707047549628, −8.165314308925784, −7.529630962036656, −7.145416547704956, −6.720848455171089, −5.992561278276126, −5.687341236814878, −4.970959772206168, −4.347843614812879, −3.840227531681578, −3.373161088009611, −3.066492319152254, −2.431902061359897, −1.834872869800868, −1.292402248720998, 0, 0,
1.292402248720998, 1.834872869800868, 2.431902061359897, 3.066492319152254, 3.373161088009611, 3.840227531681578, 4.347843614812879, 4.970959772206168, 5.687341236814878, 5.992561278276126, 6.720848455171089, 7.145416547704956, 7.529630962036656, 8.165314308925784, 8.433707047549628, 8.960582647131557, 9.256862598057485, 9.856576056358523, 10.21623067314784, 10.72398929553510, 11.30218552940822, 11.66187595207706, 12.33346347133028, 12.73524292217642, 12.95802206387801
