| L(s) = 1 | + 0.715·2-s + 3.10·3-s − 1.48·4-s + 0.799·5-s + 2.22·6-s − 0.891·7-s − 2.49·8-s + 6.67·9-s + 0.572·10-s + 0.790·11-s − 4.62·12-s + 6.94·13-s − 0.638·14-s + 2.48·15-s + 1.18·16-s + 17-s + 4.77·18-s − 0.850·19-s − 1.18·20-s − 2.77·21-s + 0.565·22-s − 3.83·23-s − 7.76·24-s − 4.36·25-s + 4.97·26-s + 11.4·27-s + 1.32·28-s + ⋯ |
| L(s) = 1 | + 0.506·2-s + 1.79·3-s − 0.743·4-s + 0.357·5-s + 0.908·6-s − 0.336·7-s − 0.882·8-s + 2.22·9-s + 0.180·10-s + 0.238·11-s − 1.33·12-s + 1.92·13-s − 0.170·14-s + 0.642·15-s + 0.297·16-s + 0.242·17-s + 1.12·18-s − 0.195·19-s − 0.265·20-s − 0.604·21-s + 0.120·22-s − 0.799·23-s − 1.58·24-s − 0.872·25-s + 0.975·26-s + 2.19·27-s + 0.250·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.396866134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.396866134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 0.715T + 2T^{2} \) |
| 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 - 0.799T + 5T^{2} \) |
| 7 | \( 1 + 0.891T + 7T^{2} \) |
| 11 | \( 1 - 0.790T + 11T^{2} \) |
| 13 | \( 1 - 6.94T + 13T^{2} \) |
| 19 | \( 1 + 0.850T + 19T^{2} \) |
| 23 | \( 1 + 3.83T + 23T^{2} \) |
| 29 | \( 1 - 2.63T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 0.234T + 37T^{2} \) |
| 41 | \( 1 - 5.91T + 41T^{2} \) |
| 43 | \( 1 - 0.280T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 - 4.91T + 53T^{2} \) |
| 61 | \( 1 - 7.96T + 61T^{2} \) |
| 67 | \( 1 + 9.09T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 4.12T + 73T^{2} \) |
| 79 | \( 1 + 4.06T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.677950965667636661974189105731, −9.044723779695130323015226980649, −8.466941672558553650981228223746, −7.79819778854635117401126873628, −6.49836224584048497680659159030, −5.67278115318728060454862807703, −4.17346066316517334277533842077, −3.75307156385294853993908872606, −2.85285121376917271332321967218, −1.50965200141838347296684505289,
1.50965200141838347296684505289, 2.85285121376917271332321967218, 3.75307156385294853993908872606, 4.17346066316517334277533842077, 5.67278115318728060454862807703, 6.49836224584048497680659159030, 7.79819778854635117401126873628, 8.466941672558553650981228223746, 9.044723779695130323015226980649, 9.677950965667636661974189105731
