| L(s) = 1 | − 3·5-s + 3·7-s − 3·9-s + 3·11-s − 13-s − 7·17-s + 19-s + 4·25-s − 8·29-s + 2·31-s − 9·35-s − 8·37-s + 2·41-s + 43-s + 9·45-s − 7·47-s + 2·49-s − 4·53-s − 9·55-s − 12·59-s − 9·61-s − 9·63-s + 3·65-s + 2·67-s + 4·71-s − 13·73-s + 9·77-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.13·7-s − 9-s + 0.904·11-s − 0.277·13-s − 1.69·17-s + 0.229·19-s + 4/5·25-s − 1.48·29-s + 0.359·31-s − 1.52·35-s − 1.31·37-s + 0.312·41-s + 0.152·43-s + 1.34·45-s − 1.02·47-s + 2/7·49-s − 0.549·53-s − 1.21·55-s − 1.56·59-s − 1.15·61-s − 1.13·63-s + 0.372·65-s + 0.244·67-s + 0.474·71-s − 1.52·73-s + 1.02·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 9 T + p T^{2} \) | 1.61.j |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−9.278110965789898156914370820027, −8.672005930970064790950362555157, −7.954542388357660452548542956467, −7.21967288411675099933762328896, −6.20463167086950485049149190570, −4.97133241736068208181350085176, −4.27386135815819791299466320042, −3.29112529540163574833018495075, −1.84792235775687446683121865586, 0,
1.84792235775687446683121865586, 3.29112529540163574833018495075, 4.27386135815819791299466320042, 4.97133241736068208181350085176, 6.20463167086950485049149190570, 7.21967288411675099933762328896, 7.954542388357660452548542956467, 8.672005930970064790950362555157, 9.278110965789898156914370820027
