| L(s) = 1 | + 3·3-s + 5-s + 7-s + 6·9-s + 6·11-s − 13-s + 3·15-s − 17-s + 5·19-s + 3·21-s + 6·23-s + 25-s + 9·27-s − 9·29-s + 31-s + 18·33-s + 35-s − 6·37-s − 3·39-s − 2·41-s − 8·43-s + 6·45-s + 5·47-s + 49-s − 3·51-s − 13·53-s + 6·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s + 1.80·11-s − 0.277·13-s + 0.774·15-s − 0.242·17-s + 1.14·19-s + 0.654·21-s + 1.25·23-s + 1/5·25-s + 1.73·27-s − 1.67·29-s + 0.179·31-s + 3.13·33-s + 0.169·35-s − 0.986·37-s − 0.480·39-s − 0.312·41-s − 1.21·43-s + 0.894·45-s + 0.729·47-s + 1/7·49-s − 0.420·51-s − 1.78·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.033487144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.033487144\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−8.486271809744697755804468046703, −7.51574006180340724386541302098, −7.13923390305044343772136610789, −6.31046801799870618115267353819, −5.20089518767868451381609349484, −4.37106058411736640352693069634, −3.52170145300683028368215221425, −3.02297263612137677022366957495, −1.84004660972837261878261654597, −1.37343561648678528291507794497,
1.37343561648678528291507794497, 1.84004660972837261878261654597, 3.02297263612137677022366957495, 3.52170145300683028368215221425, 4.37106058411736640352693069634, 5.20089518767868451381609349484, 6.31046801799870618115267353819, 7.13923390305044343772136610789, 7.51574006180340724386541302098, 8.486271809744697755804468046703
