L(s) = 1 | + 2-s + 3-s + 4-s + 0.245·5-s + 6-s + 7-s + 8-s + 9-s + 0.245·10-s − 0.943·11-s + 12-s − 0.367·13-s + 14-s + 0.245·15-s + 16-s − 2.51·17-s + 18-s + 1.60·19-s + 0.245·20-s + 21-s − 0.943·22-s + 3.95·23-s + 24-s − 4.93·25-s − 0.367·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.109·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.0777·10-s − 0.284·11-s + 0.288·12-s − 0.101·13-s + 0.267·14-s + 0.0634·15-s + 0.250·16-s − 0.610·17-s + 0.235·18-s + 0.367·19-s + 0.0549·20-s + 0.218·21-s − 0.201·22-s + 0.824·23-s + 0.204·24-s − 0.987·25-s − 0.0720·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.610675912\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.610675912\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 191 | \( 1 + T \) |
good | 5 | \( 1 - 0.245T + 5T^{2} \) |
| 11 | \( 1 + 0.943T + 11T^{2} \) |
| 13 | \( 1 + 0.367T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 - 1.60T + 19T^{2} \) |
| 23 | \( 1 - 3.95T + 23T^{2} \) |
| 29 | \( 1 - 7.83T + 29T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 2.94T + 41T^{2} \) |
| 43 | \( 1 + 5.73T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 - 4.26T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 2.86T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 2.56T + 73T^{2} \) |
| 79 | \( 1 + 1.82T + 79T^{2} \) |
| 83 | \( 1 - 1.89T + 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
| 97 | \( 1 + 5.30T + 97T^{2} \) |
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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
−7.88008810335130382948858837278, −6.99626984609600367861301116729, −6.57144264100428783237806066067, −5.57651708009953706033202750365, −4.99173757141941671521799703746, −4.28038559614032026237238153691, −3.54568684805741900437636000668, −2.67132491855178969580913446033, −2.08829218977500218287691566091, −0.948651996566829852915197434754,
0.948651996566829852915197434754, 2.08829218977500218287691566091, 2.67132491855178969580913446033, 3.54568684805741900437636000668, 4.28038559614032026237238153691, 4.99173757141941671521799703746, 5.57651708009953706033202750365, 6.57144264100428783237806066067, 6.99626984609600367861301116729, 7.88008810335130382948858837278