| L(s) = 1 | − 3·3-s − 5-s + 3·7-s + 6·9-s − 4·11-s + 6·13-s + 3·15-s − 6·17-s − 7·19-s − 9·21-s − 6·23-s − 4·25-s − 9·27-s − 3·29-s + 8·31-s + 12·33-s − 3·35-s + 2·37-s − 18·39-s + 3·41-s − 12·43-s − 6·45-s − 2·47-s + 2·49-s + 18·51-s − 5·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 1.13·7-s + 2·9-s − 1.20·11-s + 1.66·13-s + 0.774·15-s − 1.45·17-s − 1.60·19-s − 1.96·21-s − 1.25·23-s − 4/5·25-s − 1.73·27-s − 0.557·29-s + 1.43·31-s + 2.08·33-s − 0.507·35-s + 0.328·37-s − 2.88·39-s + 0.468·41-s − 1.82·43-s − 0.894·45-s − 0.291·47-s + 2/7·49-s + 2.52·51-s − 0.686·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 59 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−10.94658929455733311587618746575, −10.16325801611112073245916678153, −8.492333631755812106654761049482, −7.900945059331920291866101556024, −6.51606669659105178639024490851, −5.93847996182665973555132905783, −4.78559922440218031645291326399, −4.16126891764796093870334598032, −1.82468603249805146799859685497, 0,
1.82468603249805146799859685497, 4.16126891764796093870334598032, 4.78559922440218031645291326399, 5.93847996182665973555132905783, 6.51606669659105178639024490851, 7.900945059331920291866101556024, 8.492333631755812106654761049482, 10.16325801611112073245916678153, 10.94658929455733311587618746575
