| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s − 7-s + 8-s + 9-s − 2·10-s − 12-s − 13-s − 14-s + 2·15-s + 16-s + 6·17-s + 18-s + 2·19-s − 2·20-s + 21-s + 4·23-s − 24-s − 25-s − 26-s − 27-s − 28-s + 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s − 0.447·20-s + 0.218·21-s + 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.873766820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.873766820\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.10497228100244, −13.83524810991673, −12.95367782465924, −12.73269915973907, −12.11611383955783, −11.72676576518328, −11.52646037829633, −10.63238361576527, −10.34130543951263, −9.815437912402234, −9.086626485096096, −8.503712430336960, −7.748168172211443, −7.365259570472556, −7.014981081126223, −6.223273262182884, −5.689038717886698, −5.301323215621111, −4.514051128135217, −4.173929416126607, −3.342805435431653, −3.068478605808194, −2.189776997662100, −1.138834008052376, −0.6036753779825330,
0.6036753779825330, 1.138834008052376, 2.189776997662100, 3.068478605808194, 3.342805435431653, 4.173929416126607, 4.514051128135217, 5.301323215621111, 5.689038717886698, 6.223273262182884, 7.014981081126223, 7.365259570472556, 7.748168172211443, 8.503712430336960, 9.086626485096096, 9.815437912402234, 10.34130543951263, 10.63238361576527, 11.52646037829633, 11.72676576518328, 12.11611383955783, 12.73269915973907, 12.95367782465924, 13.83524810991673, 14.10497228100244
