| L(s) = 1 | + 2.14·2-s − 3-s + 2.61·4-s − 3.62·5-s − 2.14·6-s + 2.48·7-s + 1.31·8-s + 9-s − 7.77·10-s + 11-s − 2.61·12-s + 0.181·13-s + 5.34·14-s + 3.62·15-s − 2.40·16-s − 1.02·17-s + 2.14·18-s + 8.45·19-s − 9.45·20-s − 2.48·21-s + 2.14·22-s − 4.42·23-s − 1.31·24-s + 8.11·25-s + 0.390·26-s − 27-s + 6.49·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 0.577·3-s + 1.30·4-s − 1.61·5-s − 0.876·6-s + 0.940·7-s + 0.463·8-s + 0.333·9-s − 2.45·10-s + 0.301·11-s − 0.753·12-s + 0.0504·13-s + 1.42·14-s + 0.935·15-s − 0.601·16-s − 0.248·17-s + 0.506·18-s + 1.93·19-s − 2.11·20-s − 0.542·21-s + 0.457·22-s − 0.923·23-s − 0.267·24-s + 1.62·25-s + 0.0765·26-s − 0.192·27-s + 1.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.826703262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.826703262\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 13 | \( 1 - 0.181T + 13T^{2} \) |
| 17 | \( 1 + 1.02T + 17T^{2} \) |
| 19 | \( 1 - 8.45T + 19T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 - 5.84T + 29T^{2} \) |
| 31 | \( 1 - 9.14T + 31T^{2} \) |
| 37 | \( 1 - 1.08T + 37T^{2} \) |
| 41 | \( 1 - 6.75T + 41T^{2} \) |
| 43 | \( 1 - 3.54T + 43T^{2} \) |
| 47 | \( 1 - 3.99T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 8.20T + 73T^{2} \) |
| 79 | \( 1 - 5.35T + 79T^{2} \) |
| 83 | \( 1 - 9.24T + 83T^{2} \) |
| 89 | \( 1 + 8.09T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−9.017650927635092055587314442256, −8.004012988455660122802414982396, −7.49804518212729584678541135668, −6.62560272046312071771246209379, −5.73464602980019158978251560356, −4.85205192827043950505695726067, −4.37785367130210513884153469795, −3.67403455678366703584061079133, −2.68031886695995147392289966265, −0.954310657400618182631338079836,
0.954310657400618182631338079836, 2.68031886695995147392289966265, 3.67403455678366703584061079133, 4.37785367130210513884153469795, 4.85205192827043950505695726067, 5.73464602980019158978251560356, 6.62560272046312071771246209379, 7.49804518212729584678541135668, 8.004012988455660122802414982396, 9.017650927635092055587314442256
