| L(s) = 1 | + 2-s − 1.60·3-s + 4-s + 1.46·5-s − 1.60·6-s + 4.50·7-s + 8-s − 0.427·9-s + 1.46·10-s + 0.640·11-s − 1.60·12-s − 0.795·13-s + 4.50·14-s − 2.34·15-s + 16-s + 3.01·17-s − 0.427·18-s − 4.80·19-s + 1.46·20-s − 7.22·21-s + 0.640·22-s + 4.47·23-s − 1.60·24-s − 2.85·25-s − 0.795·26-s + 5.49·27-s + 4.50·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.926·3-s + 0.5·4-s + 0.654·5-s − 0.654·6-s + 1.70·7-s + 0.353·8-s − 0.142·9-s + 0.462·10-s + 0.193·11-s − 0.463·12-s − 0.220·13-s + 1.20·14-s − 0.606·15-s + 0.250·16-s + 0.732·17-s − 0.100·18-s − 1.10·19-s + 0.327·20-s − 1.57·21-s + 0.136·22-s + 0.933·23-s − 0.327·24-s − 0.571·25-s − 0.155·26-s + 1.05·27-s + 0.850·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.119425671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.119425671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 1.60T + 3T^{2} \) |
| 5 | \( 1 - 1.46T + 5T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 11 | \( 1 - 0.640T + 11T^{2} \) |
| 13 | \( 1 + 0.795T + 13T^{2} \) |
| 17 | \( 1 - 3.01T + 17T^{2} \) |
| 19 | \( 1 + 4.80T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 - 4.51T + 29T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 + 5.08T + 43T^{2} \) |
| 47 | \( 1 + 9.95T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 0.0329T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 6.30T + 89T^{2} \) |
| 97 | \( 1 + 7.18T + 97T^{2} \) |
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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
−8.106634604843543953004066856123, −7.85761927325924102493414382306, −6.48913158642426407465247725210, −6.27168064740029995151135433974, −5.19765975108989065874650279439, −4.98102805737806453580218542063, −4.20005086903382009310255096906, −2.88738372218109684172544739252, −1.95191814193546156796625762842, −1.01796481492197130474140608672,
1.01796481492197130474140608672, 1.95191814193546156796625762842, 2.88738372218109684172544739252, 4.20005086903382009310255096906, 4.98102805737806453580218542063, 5.19765975108989065874650279439, 6.27168064740029995151135433974, 6.48913158642426407465247725210, 7.85761927325924102493414382306, 8.106634604843543953004066856123
