| L(s) = 1 | − 2-s + 3.37·3-s + 4-s − 3.37·5-s − 3.37·6-s − 8-s + 8.37·9-s + 3.37·10-s + 1.37·11-s + 3.37·12-s + 0.372·13-s − 11.3·15-s + 16-s − 4.74·17-s − 8.37·18-s − 5.37·19-s − 3.37·20-s − 1.37·22-s − 8.74·23-s − 3.37·24-s + 6.37·25-s − 0.372·26-s + 18.1·27-s − 7.11·29-s + 11.3·30-s + 2.74·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.94·3-s + 0.5·4-s − 1.50·5-s − 1.37·6-s − 0.353·8-s + 2.79·9-s + 1.06·10-s + 0.413·11-s + 0.973·12-s + 0.103·13-s − 2.93·15-s + 0.250·16-s − 1.15·17-s − 1.97·18-s − 1.23·19-s − 0.754·20-s − 0.292·22-s − 1.82·23-s − 0.688·24-s + 1.27·25-s − 0.0730·26-s + 3.48·27-s − 1.32·29-s + 2.07·30-s + 0.492·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 3.37T + 3T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 0.372T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 + 8.74T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 43 | \( 1 + 6.37T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 0.372T + 59T^{2} \) |
| 61 | \( 1 - 2.62T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 + 0.372T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.151088353727058829378633377593, −7.70416058859737732385644462708, −7.03164976737505907274552577369, −6.31305303076110485966168378273, −4.50800272966772009184168152681, −4.00947163142639427802969339758, −3.43338595660807186451950297117, −2.39318951590849486040966270499, −1.67955102688756535654849862759, 0,
1.67955102688756535654849862759, 2.39318951590849486040966270499, 3.43338595660807186451950297117, 4.00947163142639427802969339758, 4.50800272966772009184168152681, 6.31305303076110485966168378273, 7.03164976737505907274552577369, 7.70416058859737732385644462708, 8.151088353727058829378633377593
