| L(s) = 1 | + 2-s + 4-s − 5-s + 3.37·7-s + 8-s − 10-s + 11-s + 2·13-s + 3.37·14-s + 16-s − 1.37·17-s + 0.627·19-s − 20-s + 22-s − 2.74·23-s + 25-s + 2·26-s + 3.37·28-s − 1.37·29-s + 3.37·31-s + 32-s − 1.37·34-s − 3.37·35-s + 9.37·37-s + 0.627·38-s − 40-s + 11.4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.27·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s + 0.901·14-s + 0.250·16-s − 0.332·17-s + 0.144·19-s − 0.223·20-s + 0.213·22-s − 0.572·23-s + 0.200·25-s + 0.392·26-s + 0.637·28-s − 0.254·29-s + 0.605·31-s + 0.176·32-s − 0.235·34-s − 0.570·35-s + 1.54·37-s + 0.101·38-s − 0.158·40-s + 1.79·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.721182629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.721182629\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 - 9.37T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 - 2.74T + 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−10.18117660127891089939227519679, −9.033087181020552502331228861761, −8.122791105403659904684444900838, −7.55886488555460606639278772860, −6.45968307937970757559734997664, −5.58519302697905327488577221262, −4.54878270126902218209876285934, −3.98407204612482346710023372520, −2.63434882131933127885518227886, −1.34411130567678245509630805146,
1.34411130567678245509630805146, 2.63434882131933127885518227886, 3.98407204612482346710023372520, 4.54878270126902218209876285934, 5.58519302697905327488577221262, 6.45968307937970757559734997664, 7.55886488555460606639278772860, 8.122791105403659904684444900838, 9.033087181020552502331228861761, 10.18117660127891089939227519679
