| L(s) = 1 | + 2-s + 3-s + 4-s − 0.561·5-s + 6-s − 7-s + 8-s + 9-s − 0.561·10-s + 2.56·11-s + 12-s + 13-s − 14-s − 0.561·15-s + 16-s + 5.68·17-s + 18-s + 7.68·19-s − 0.561·20-s − 21-s + 2.56·22-s − 1.43·23-s + 24-s − 4.68·25-s + 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.251·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.177·10-s + 0.772·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.144·15-s + 0.250·16-s + 1.37·17-s + 0.235·18-s + 1.76·19-s − 0.125·20-s − 0.218·21-s + 0.546·22-s − 0.299·23-s + 0.204·24-s − 0.936·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.630653470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.630653470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 0.561T + 73T^{2} \) |
| 79 | \( 1 + 2.87T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−10.96102185939495039898716611437, −9.756888510714417580519514037867, −9.233370560023010947145845166715, −7.83825761795388458926986676872, −7.32618673380141590440426968804, −6.09386238561401144317148233861, −5.20605424187117081970844569536, −3.73522433295182470573736754474, −3.32218683883157689035440528794, −1.61386041996840783631232130153,
1.61386041996840783631232130153, 3.32218683883157689035440528794, 3.73522433295182470573736754474, 5.20605424187117081970844569536, 6.09386238561401144317148233861, 7.32618673380141590440426968804, 7.83825761795388458926986676872, 9.233370560023010947145845166715, 9.756888510714417580519514037867, 10.96102185939495039898716611437
