| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 2·7-s + 8-s + 9-s − 2·10-s + 11-s − 12-s + 4·13-s + 2·14-s + 2·15-s + 16-s + 18-s + 2·19-s − 2·20-s − 2·21-s + 22-s − 2·23-s − 24-s − 25-s + 4·26-s − 27-s + 2·28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 1.10·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s + 0.458·19-s − 0.447·20-s − 0.436·21-s + 0.213·22-s − 0.417·23-s − 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−15.86159791523714, −15.50305992065186, −15.03990298398650, −14.23188777649535, −13.87483198562833, −13.32382588827272, −12.55295562437688, −12.04478080242170, −11.68987157028148, −11.09389995215438, −10.82010740452089, −10.04442281646197, −9.260028655693960, −8.503739360084038, −7.890685005019509, −7.540785576904354, −6.626391647914420, −6.224244536630654, −5.479611760908804, −4.832026091366819, −4.297356253948725, −3.642322194401107, −3.090951910733605, −1.819179384530037, −1.258009721835221, 0,
1.258009721835221, 1.819179384530037, 3.090951910733605, 3.642322194401107, 4.297356253948725, 4.832026091366819, 5.479611760908804, 6.224244536630654, 6.626391647914420, 7.540785576904354, 7.890685005019509, 8.503739360084038, 9.260028655693960, 10.04442281646197, 10.82010740452089, 11.09389995215438, 11.68987157028148, 12.04478080242170, 12.55295562437688, 13.32382588827272, 13.87483198562833, 14.23188777649535, 15.03990298398650, 15.50305992065186, 15.86159791523714
