| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 2·11-s − 12-s − 13-s + 14-s + 16-s − 2·17-s + 18-s + 4·19-s − 21-s − 2·22-s − 6·23-s − 24-s − 26-s − 27-s + 28-s + 2·29-s + 4·31-s + 32-s + 2·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.218·21-s − 0.426·22-s − 1.25·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.348·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.556608965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.556608965\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−15.99740022997874, −15.66824396379056, −15.18912602603950, −14.35864067271467, −13.93359513626105, −13.43887239473343, −12.76616529626928, −12.16991241747914, −11.76722560550604, −11.22178384097971, −10.52911480712792, −10.09492262204916, −9.424672394849067, −8.497049021147223, −7.890528050968022, −7.317371888344522, −6.623279580506116, −5.984506093775411, −5.349861316574540, −4.822296070834974, −4.191537428444411, −3.382317458320354, −2.506150412332396, −1.777353750466639, −0.6480964105955423,
0.6480964105955423, 1.777353750466639, 2.506150412332396, 3.382317458320354, 4.191537428444411, 4.822296070834974, 5.349861316574540, 5.984506093775411, 6.623279580506116, 7.317371888344522, 7.890528050968022, 8.497049021147223, 9.424672394849067, 10.09492262204916, 10.52911480712792, 11.22178384097971, 11.76722560550604, 12.16991241747914, 12.76616529626928, 13.43887239473343, 13.93359513626105, 14.35864067271467, 15.18912602603950, 15.66824396379056, 15.99740022997874
