| L(s) = 1 | − 4.76·2-s + 14.7·4-s − 12.2·5-s + 22.9·7-s − 31.9·8-s + 58.3·10-s + 11.5·11-s + 51.2·13-s − 109.·14-s + 34.5·16-s + 46.1·17-s − 76.3·19-s − 180.·20-s − 54.8·22-s + 98.7·23-s + 25.1·25-s − 244.·26-s + 336.·28-s + 135.·29-s + 11.2·31-s + 90.9·32-s − 220.·34-s − 280.·35-s − 380.·37-s + 363.·38-s + 391.·40-s − 164.·41-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.83·4-s − 1.09·5-s + 1.23·7-s − 1.41·8-s + 1.84·10-s + 0.315·11-s + 1.09·13-s − 2.08·14-s + 0.539·16-s + 0.659·17-s − 0.922·19-s − 2.01·20-s − 0.531·22-s + 0.895·23-s + 0.201·25-s − 1.84·26-s + 2.27·28-s + 0.868·29-s + 0.0651·31-s + 0.502·32-s − 1.11·34-s − 1.35·35-s − 1.69·37-s + 1.55·38-s + 1.54·40-s − 0.626·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 4.76T + 8T^{2} \) |
| 5 | \( 1 + 12.2T + 125T^{2} \) |
| 7 | \( 1 - 22.9T + 343T^{2} \) |
| 11 | \( 1 - 11.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 98.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 11.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 380.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 164.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 69.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 316.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 0.636T + 2.05e5T^{2} \) |
| 61 | \( 1 - 562.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 720.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 82.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 510.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 276.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 433.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 373.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.54e3T + 9.12e5T^{2} \) |
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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
−8.423889627547866903213188268312, −7.929190052497027228173084234897, −7.10549440917049577720000595805, −6.42920744013444335760706579761, −5.13507957025008909096231704628, −4.17883182129135962526182168646, −3.16765247388532223551625789416, −1.76928483864433812237968537133, −1.11870348679528903831297710803, 0,
1.11870348679528903831297710803, 1.76928483864433812237968537133, 3.16765247388532223551625789416, 4.17883182129135962526182168646, 5.13507957025008909096231704628, 6.42920744013444335760706579761, 7.10549440917049577720000595805, 7.929190052497027228173084234897, 8.423889627547866903213188268312
