| L(s) = 1 | + 3-s − 2·5-s − 2·9-s − 3·11-s + 4·13-s − 2·15-s − 4·19-s − 7·23-s − 25-s − 5·27-s + 6·29-s − 31-s − 3·33-s − 5·37-s + 4·39-s − 12·41-s + 8·43-s + 4·45-s − 7·47-s − 7·49-s + 12·53-s + 6·55-s − 4·57-s − 5·59-s − 3·61-s − 8·65-s + 3·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 2/3·9-s − 0.904·11-s + 1.10·13-s − 0.516·15-s − 0.917·19-s − 1.45·23-s − 1/5·25-s − 0.962·27-s + 1.11·29-s − 0.179·31-s − 0.522·33-s − 0.821·37-s + 0.640·39-s − 1.87·41-s + 1.21·43-s + 0.596·45-s − 1.02·47-s − 49-s + 1.64·53-s + 0.809·55-s − 0.529·57-s − 0.650·59-s − 0.384·61-s − 0.992·65-s + 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.79183929274787, −13.55619889662388, −12.94345991652326, −12.38302689072083, −11.95620216482398, −11.49664312224692, −10.99930502970832, −10.55971807205809, −10.06308402773318, −9.540570805171112, −8.699312598176604, −8.505696046956062, −8.051872839876775, −7.858741026889326, −6.984871584780882, −6.563012254375814, −5.872476365627626, −5.515745249021467, −4.756690521987389, −4.159704289286551, −3.675985126926668, −3.249177187496170, −2.564931174360672, −2.024706414377757, −1.264647083955586, 0, 0,
1.264647083955586, 2.024706414377757, 2.564931174360672, 3.249177187496170, 3.675985126926668, 4.159704289286551, 4.756690521987389, 5.515745249021467, 5.872476365627626, 6.563012254375814, 6.984871584780882, 7.858741026889326, 8.051872839876775, 8.505696046956062, 8.699312598176604, 9.540570805171112, 10.06308402773318, 10.55971807205809, 10.99930502970832, 11.49664312224692, 11.95620216482398, 12.38302689072083, 12.94345991652326, 13.55619889662388, 13.79183929274787
