| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 2·7-s − 2·9-s + 2·10-s − 11-s − 2·12-s − 4·13-s + 4·14-s + 15-s − 4·16-s − 2·17-s + 4·18-s − 2·20-s + 2·21-s + 2·22-s + 23-s − 4·25-s + 8·26-s + 5·27-s − 4·28-s − 2·30-s − 7·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 2/3·9-s + 0.632·10-s − 0.301·11-s − 0.577·12-s − 1.10·13-s + 1.06·14-s + 0.258·15-s − 16-s − 0.485·17-s + 0.942·18-s − 0.447·20-s + 0.436·21-s + 0.426·22-s + 0.208·23-s − 4/5·25-s + 1.56·26-s + 0.962·27-s − 0.755·28-s − 0.365·30-s − 1.25·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) | |
| 47 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.10866115166621, −15.86402747979300, −14.93795016241798, −14.50868232960937, −13.83186429415074, −12.97856416476612, −12.71564505031377, −11.92380825104078, −11.39668303675714, −11.01906627443928, −10.42850664323642, −9.844519506423441, −9.314430896032731, −8.973496041863978, −8.081462103029910, −7.798153654693855, −7.058207941210378, −6.659194761809159, −5.831040557478353, −5.273889914667727, −4.464794641652434, −3.763535882342020, −2.705327597986168, −2.293290479493529, −1.108497976363629, 0, 0,
1.108497976363629, 2.293290479493529, 2.705327597986168, 3.763535882342020, 4.464794641652434, 5.273889914667727, 5.831040557478353, 6.659194761809159, 7.058207941210378, 7.798153654693855, 8.081462103029910, 8.973496041863978, 9.314430896032731, 9.844519506423441, 10.42850664323642, 11.01906627443928, 11.39668303675714, 11.92380825104078, 12.71564505031377, 12.97856416476612, 13.83186429415074, 14.50868232960937, 14.93795016241798, 15.86402747979300, 16.10866115166621
