| L(s) = 1 | − 3-s − 7-s + 9-s − 13-s + 7·19-s + 21-s − 8·23-s − 5·25-s − 27-s − 8·29-s + 31-s − 3·37-s + 39-s + 8·41-s − 8·43-s − 12·47-s − 6·49-s − 6·53-s − 7·57-s − 6·59-s − 61-s − 63-s + 13·67-s + 8·69-s − 12·71-s − 73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 1.60·19-s + 0.218·21-s − 1.66·23-s − 25-s − 0.192·27-s − 1.48·29-s + 0.179·31-s − 0.493·37-s + 0.160·39-s + 1.24·41-s − 1.21·43-s − 1.75·47-s − 6/7·49-s − 0.824·53-s − 0.927·57-s − 0.781·59-s − 0.128·61-s − 0.125·63-s + 1.58·67-s + 0.963·69-s − 1.42·71-s − 0.117·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
| 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
−13.69435218842330, −13.34834259826734, −12.81244957519362, −12.36125619358027, −11.71726731966860, −11.62098945596338, −11.06777823805536, −10.42676274191130, −9.767964669917890, −9.710938921876508, −9.256293932895967, −8.346208204091868, −7.907930702754258, −7.506642342967522, −6.991533832370942, −6.254974346957170, −6.033087882577113, −5.356945813686001, −5.004183276923176, −4.277493008156697, −3.670828048848827, −3.293957717405554, −2.486467105793706, −1.735459131241282, −1.294099659429168, 0, 0,
1.294099659429168, 1.735459131241282, 2.486467105793706, 3.293957717405554, 3.670828048848827, 4.277493008156697, 5.004183276923176, 5.356945813686001, 6.033087882577113, 6.254974346957170, 6.991533832370942, 7.506642342967522, 7.907930702754258, 8.346208204091868, 9.256293932895967, 9.710938921876508, 9.767964669917890, 10.42676274191130, 11.06777823805536, 11.62098945596338, 11.71726731966860, 12.36125619358027, 12.81244957519362, 13.34834259826734, 13.69435218842330
