| L(s) = 1 | + 2-s + 2.90·3-s + 4-s − 4.24·5-s + 2.90·6-s + 1.64·7-s + 8-s + 5.44·9-s − 4.24·10-s − 4.42·11-s + 2.90·12-s + 5.19·13-s + 1.64·14-s − 12.3·15-s + 16-s − 2.76·17-s + 5.44·18-s + 3.96·19-s − 4.24·20-s + 4.79·21-s − 4.42·22-s − 1.54·23-s + 2.90·24-s + 13.0·25-s + 5.19·26-s + 7.09·27-s + 1.64·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.67·3-s + 0.5·4-s − 1.89·5-s + 1.18·6-s + 0.623·7-s + 0.353·8-s + 1.81·9-s − 1.34·10-s − 1.33·11-s + 0.838·12-s + 1.44·13-s + 0.440·14-s − 3.18·15-s + 0.250·16-s − 0.671·17-s + 1.28·18-s + 0.908·19-s − 0.949·20-s + 1.04·21-s − 0.943·22-s − 0.321·23-s + 0.593·24-s + 2.60·25-s + 1.01·26-s + 1.36·27-s + 0.311·28-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
| \(L(1)\) |
\(\approx\) |
\(4.459915041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.459915041\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 + 4.42T + 11T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 - 3.96T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 - 6.50T + 29T^{2} \) |
| 31 | \( 1 - 5.30T + 31T^{2} \) |
| 37 | \( 1 - 2.65T + 37T^{2} \) |
| 41 | \( 1 - 9.15T + 41T^{2} \) |
| 43 | \( 1 + 0.827T + 43T^{2} \) |
| 47 | \( 1 + 6.25T + 47T^{2} \) |
| 53 | \( 1 - 4.91T + 53T^{2} \) |
| 59 | \( 1 - 4.62T + 59T^{2} \) |
| 61 | \( 1 - 7.56T + 61T^{2} \) |
| 67 | \( 1 + 3.26T + 67T^{2} \) |
| 71 | \( 1 - 3.04T + 71T^{2} \) |
| 73 | \( 1 + 7.11T + 73T^{2} \) |
| 79 | \( 1 - 2.43T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 6.76T + 97T^{2} \) |
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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
−8.350571851240211038437446553563, −7.82645467905777757521065369813, −7.34893616293728826387033615603, −6.40744645383697712890447260207, −5.06147046865646564979928323184, −4.38063700272096573039859649844, −3.78448145702734784764888908612, −3.09436161343641213222743557071, −2.46789357482591161505032841705, −1.05465405740400052315182665113,
1.05465405740400052315182665113, 2.46789357482591161505032841705, 3.09436161343641213222743557071, 3.78448145702734784764888908612, 4.38063700272096573039859649844, 5.06147046865646564979928323184, 6.40744645383697712890447260207, 7.34893616293728826387033615603, 7.82645467905777757521065369813, 8.350571851240211038437446553563
