| L(s) = 1 | − 0.791·2-s + 0.403·3-s − 1.37·4-s + 5-s − 0.318·6-s + 4.95·7-s + 2.66·8-s − 2.83·9-s − 0.791·10-s + 1.72·11-s − 0.553·12-s − 6.32·13-s − 3.91·14-s + 0.403·15-s + 0.636·16-s + 4.42·17-s + 2.24·18-s + 1.59·19-s − 1.37·20-s + 1.99·21-s − 1.36·22-s + 3.09·23-s + 1.07·24-s + 25-s + 5.00·26-s − 2.35·27-s − 6.80·28-s + ⋯ |
| L(s) = 1 | − 0.559·2-s + 0.232·3-s − 0.687·4-s + 0.447·5-s − 0.130·6-s + 1.87·7-s + 0.943·8-s − 0.945·9-s − 0.250·10-s + 0.520·11-s − 0.159·12-s − 1.75·13-s − 1.04·14-s + 0.104·15-s + 0.159·16-s + 1.07·17-s + 0.529·18-s + 0.365·19-s − 0.307·20-s + 0.435·21-s − 0.291·22-s + 0.646·23-s + 0.219·24-s + 0.200·25-s + 0.981·26-s − 0.452·27-s − 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.532860624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.532860624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.791T + 2T^{2} \) |
| 3 | \( 1 - 0.403T + 3T^{2} \) |
| 7 | \( 1 - 4.95T + 7T^{2} \) |
| 11 | \( 1 - 1.72T + 11T^{2} \) |
| 13 | \( 1 + 6.32T + 13T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 - 3.59T + 29T^{2} \) |
| 31 | \( 1 - 6.53T + 31T^{2} \) |
| 37 | \( 1 - 8.41T + 37T^{2} \) |
| 41 | \( 1 + 9.81T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 - 0.745T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 0.447T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 - 5.13T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 6.32T + 83T^{2} \) |
| 89 | \( 1 + 0.806T + 89T^{2} \) |
| 97 | \( 1 + 2.17T + 97T^{2} \) |
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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
−9.072483436902034594517394966578, −8.334596921828541615977104302002, −7.905690565088745357032391851794, −7.15109825412768767735360402066, −5.77207447670319057308333925088, −4.88964103881797238040768406035, −4.67062113840036448736992348696, −3.13098256842439973819833935202, −1.99980824808621649642917922892, −0.947072662004089733387614890041,
0.947072662004089733387614890041, 1.99980824808621649642917922892, 3.13098256842439973819833935202, 4.67062113840036448736992348696, 4.88964103881797238040768406035, 5.77207447670319057308333925088, 7.15109825412768767735360402066, 7.905690565088745357032391851794, 8.334596921828541615977104302002, 9.072483436902034594517394966578
