| L(s) = 1 | + 2-s − 0.732·3-s + 4-s − 1.73·5-s − 0.732·6-s + 2·7-s + 8-s − 2.46·9-s − 1.73·10-s − 1.73·11-s − 0.732·12-s + 0.732·13-s + 2·14-s + 1.26·15-s + 16-s + 1.73·17-s − 2.46·18-s + 6.73·19-s − 1.73·20-s − 1.46·21-s − 1.73·22-s − 23-s − 0.732·24-s − 2.00·25-s + 0.732·26-s + 4·27-s + 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.422·3-s + 0.5·4-s − 0.774·5-s − 0.298·6-s + 0.755·7-s + 0.353·8-s − 0.821·9-s − 0.547·10-s − 0.522·11-s − 0.211·12-s + 0.203·13-s + 0.534·14-s + 0.327·15-s + 0.250·16-s + 0.420·17-s − 0.580·18-s + 1.54·19-s − 0.387·20-s − 0.319·21-s − 0.369·22-s − 0.208·23-s − 0.149·24-s − 0.400·25-s + 0.143·26-s + 0.769·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192242357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192242357\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 0.732T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 + 0.464T + 41T^{2} \) |
| 43 | \( 1 + 9.19T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 - 7.26T + 59T^{2} \) |
| 61 | \( 1 + 3.19T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 7.26T + 71T^{2} \) |
| 73 | \( 1 - 4.19T + 73T^{2} \) |
| 79 | \( 1 + 6.53T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 + 0.535T + 97T^{2} \) |
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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
−7.933506902236091794358774228026, −7.45413960591466768163952019674, −6.57852349605830215193698653946, −5.58798529720733846884818179485, −5.35934681291178715900832323082, −4.53128499341987000471174941475, −3.64607699838655699789428732348, −3.02714417335520480234224273755, −1.95291978279713526175550188647, −0.70871907344883805469090508383,
0.70871907344883805469090508383, 1.95291978279713526175550188647, 3.02714417335520480234224273755, 3.64607699838655699789428732348, 4.53128499341987000471174941475, 5.35934681291178715900832323082, 5.58798529720733846884818179485, 6.57852349605830215193698653946, 7.45413960591466768163952019674, 7.933506902236091794358774228026
