| L(s) = 1 | + 5.43·2-s + 5.22·3-s + 21.5·4-s − 10.5·5-s + 28.3·6-s − 7·7-s + 73.5·8-s + 0.305·9-s − 57.2·10-s + 64.6·11-s + 112.·12-s + 66.6·13-s − 38.0·14-s − 55.0·15-s + 227.·16-s + 1.66·18-s + 34.1·19-s − 226.·20-s − 36.5·21-s + 351.·22-s − 180.·23-s + 384.·24-s − 13.9·25-s + 362.·26-s − 139.·27-s − 150.·28-s + 85.0·29-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 1.00·3-s + 2.69·4-s − 0.942·5-s + 1.93·6-s − 0.377·7-s + 3.25·8-s + 0.0113·9-s − 1.81·10-s + 1.77·11-s + 2.70·12-s + 1.42·13-s − 0.726·14-s − 0.947·15-s + 3.55·16-s + 0.0217·18-s + 0.412·19-s − 2.53·20-s − 0.380·21-s + 3.40·22-s − 1.63·23-s + 3.26·24-s − 0.111·25-s + 2.73·26-s − 0.994·27-s − 1.01·28-s + 0.544·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.62059133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.62059133\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 5.43T + 8T^{2} \) |
| 3 | \( 1 - 5.22T + 27T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 11 | \( 1 - 64.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.6T + 2.19e3T^{2} \) |
| 19 | \( 1 - 34.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 85.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 247.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 148.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 186.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 199.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 425.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 551.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 558.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 490.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 341.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 932.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 96.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 191.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 21.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 779.T + 9.12e5T^{2} \) |
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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
−8.511011889258174047745745648951, −7.910881544663763146597060991903, −6.94314239534185442350136196666, −6.30106571162376491848094590204, −5.62154380124293996901866486860, −4.13120566693229702583536663281, −3.93579427628830303316085561246, −3.36407493701553231503243181397, −2.35974836367974746680840193388, −1.23455020828101574548823215203,
1.23455020828101574548823215203, 2.35974836367974746680840193388, 3.36407493701553231503243181397, 3.93579427628830303316085561246, 4.13120566693229702583536663281, 5.62154380124293996901866486860, 6.30106571162376491848094590204, 6.94314239534185442350136196666, 7.910881544663763146597060991903, 8.511011889258174047745745648951
