| L(s) = 1 | + 2-s − 4-s + 2·5-s + 5·7-s − 3·8-s + 2·10-s + 2·13-s + 5·14-s − 16-s + 7·17-s − 2·20-s + 23-s − 25-s + 2·26-s − 5·28-s + 3·29-s − 8·31-s + 5·32-s + 7·34-s + 10·35-s − 3·37-s − 6·40-s − 11·41-s + 9·43-s + 46-s + 47-s + 18·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.88·7-s − 1.06·8-s + 0.632·10-s + 0.554·13-s + 1.33·14-s − 1/4·16-s + 1.69·17-s − 0.447·20-s + 0.208·23-s − 1/5·25-s + 0.392·26-s − 0.944·28-s + 0.557·29-s − 1.43·31-s + 0.883·32-s + 1.20·34-s + 1.69·35-s − 0.493·37-s − 0.948·40-s − 1.71·41-s + 1.37·43-s + 0.147·46-s + 0.145·47-s + 18/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.539609490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.539609490\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−8.659697116337032624308839590000, −7.968891492177536985677332440798, −7.20640242911520364864011440722, −5.89515644588746914285908151546, −5.54506843846831168788229741603, −4.93586054640943580460654297879, −4.10453433686104879502891739715, −3.21944769367031592181749892064, −1.98345181840166442553466640278, −1.13959427830355409302245995745,
1.13959427830355409302245995745, 1.98345181840166442553466640278, 3.21944769367031592181749892064, 4.10453433686104879502891739715, 4.93586054640943580460654297879, 5.54506843846831168788229741603, 5.89515644588746914285908151546, 7.20640242911520364864011440722, 7.968891492177536985677332440798, 8.659697116337032624308839590000
