| L(s) = 1 | + 2-s − 2·3-s + 4-s + 2·5-s − 2·6-s + 4·7-s + 8-s + 9-s + 2·10-s − 6·11-s − 2·12-s − 13-s + 4·14-s − 4·15-s + 16-s + 4·17-s + 18-s − 6·19-s + 2·20-s − 8·21-s − 6·22-s − 4·23-s − 2·24-s − 25-s − 26-s + 4·27-s + 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.80·11-s − 0.577·12-s − 0.277·13-s + 1.06·14-s − 1.03·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.37·19-s + 0.447·20-s − 1.74·21-s − 1.27·22-s − 0.834·23-s − 0.408·24-s − 1/5·25-s − 0.196·26-s + 0.769·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.355250188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.355250188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−14.60530544182477, −14.12776503234797, −13.71030834812625, −13.03839715568153, −12.63243368882646, −11.99160002003566, −11.76433296025945, −10.86847987579993, −10.64238450617760, −10.39063277192072, −9.681382817286194, −8.748042625018544, −8.060866546248831, −7.824350108922501, −7.058282696448415, −6.299217274159130, −5.813393582419786, −5.362832312820536, −5.011932557046913, −4.568110155586443, −3.698459307039581, −2.683383902888164, −2.132827625892885, −1.604709478138374, −0.5023192753983633,
0.5023192753983633, 1.604709478138374, 2.132827625892885, 2.683383902888164, 3.698459307039581, 4.568110155586443, 5.011932557046913, 5.362832312820536, 5.813393582419786, 6.299217274159130, 7.058282696448415, 7.824350108922501, 8.060866546248831, 8.748042625018544, 9.681382817286194, 10.39063277192072, 10.64238450617760, 10.86847987579993, 11.76433296025945, 11.99160002003566, 12.63243368882646, 13.03839715568153, 13.71030834812625, 14.12776503234797, 14.60530544182477
