| L(s) = 1 | − 3-s + 9-s − 2·13-s − 7·17-s + 2·19-s − 4·23-s − 5·25-s − 27-s + 8·29-s − 31-s − 2·37-s + 2·39-s + 9·41-s + 9·47-s + 7·51-s − 6·53-s − 2·57-s + 2·61-s + 8·67-s + 4·69-s − 3·71-s − 3·73-s + 5·75-s + 3·79-s + 81-s + 6·83-s − 8·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.554·13-s − 1.69·17-s + 0.458·19-s − 0.834·23-s − 25-s − 0.192·27-s + 1.48·29-s − 0.179·31-s − 0.328·37-s + 0.320·39-s + 1.40·41-s + 1.31·47-s + 0.980·51-s − 0.824·53-s − 0.264·57-s + 0.256·61-s + 0.977·67-s + 0.481·69-s − 0.356·71-s − 0.351·73-s + 0.577·75-s + 0.337·79-s + 1/9·81-s + 0.658·83-s − 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.55924514110016, −13.22133459948451, −12.49017837340871, −12.24146196650364, −11.75911408335740, −11.15409664086226, −10.93091668681135, −10.19880952218783, −9.931694705729954, −9.256042312992353, −8.919106641133482, −8.199100533139898, −7.776640327249370, −7.177670626220363, −6.727180082017896, −6.170709822871603, −5.786754124503607, −5.097295392347310, −4.587710503392708, −4.152262353045627, −3.595587415498809, −2.612254507864268, −2.322098603860368, −1.546821240368716, −0.6989080903730516, 0,
0.6989080903730516, 1.546821240368716, 2.322098603860368, 2.612254507864268, 3.595587415498809, 4.152262353045627, 4.587710503392708, 5.097295392347310, 5.786754124503607, 6.170709822871603, 6.727180082017896, 7.177670626220363, 7.776640327249370, 8.199100533139898, 8.919106641133482, 9.256042312992353, 9.931694705729954, 10.19880952218783, 10.93091668681135, 11.15409664086226, 11.75911408335740, 12.24146196650364, 12.49017837340871, 13.22133459948451, 13.55924514110016
