| L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s + 4·7-s + 8-s − 2·9-s − 3·10-s + 11-s − 12-s + 13-s + 4·14-s + 3·15-s + 16-s + 4·17-s − 2·18-s − 3·20-s − 4·21-s + 22-s − 8·23-s − 24-s + 4·25-s + 26-s + 5·27-s + 4·28-s + 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s + 0.774·15-s + 1/4·16-s + 0.970·17-s − 0.471·18-s − 0.670·20-s − 0.872·21-s + 0.213·22-s − 1.66·23-s − 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.962·27-s + 0.755·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.910544976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910544976\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 53 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−10.12481027514319639529583429234, −8.503708911464607237269363055524, −8.122543810174516103082005390560, −7.38490358491497917384950705072, −6.22310743247048972966292776202, −5.44310089052417322920645423969, −4.50802843366227304014367980399, −3.94290248055525620838713430634, −2.64712020212556420622598119491, −1.01517801401577255216547014058,
1.01517801401577255216547014058, 2.64712020212556420622598119491, 3.94290248055525620838713430634, 4.50802843366227304014367980399, 5.44310089052417322920645423969, 6.22310743247048972966292776202, 7.38490358491497917384950705072, 8.122543810174516103082005390560, 8.503708911464607237269363055524, 10.12481027514319639529583429234
