| L(s) = 1 | + 2-s + 3.15·3-s + 4-s − 3.45·5-s + 3.15·6-s + 2.97·7-s + 8-s + 6.96·9-s − 3.45·10-s + 5.83·11-s + 3.15·12-s + 0.490·13-s + 2.97·14-s − 10.9·15-s + 16-s + 1.66·17-s + 6.96·18-s − 5.00·19-s − 3.45·20-s + 9.40·21-s + 5.83·22-s + 4.18·23-s + 3.15·24-s + 6.94·25-s + 0.490·26-s + 12.5·27-s + 2.97·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.82·3-s + 0.5·4-s − 1.54·5-s + 1.28·6-s + 1.12·7-s + 0.353·8-s + 2.32·9-s − 1.09·10-s + 1.76·11-s + 0.911·12-s + 0.135·13-s + 0.795·14-s − 2.81·15-s + 0.250·16-s + 0.404·17-s + 1.64·18-s − 1.14·19-s − 0.772·20-s + 2.05·21-s + 1.24·22-s + 0.872·23-s + 0.644·24-s + 1.38·25-s + 0.0961·26-s + 2.40·27-s + 0.562·28-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
| \(L(1)\) |
\(\approx\) |
\(5.879526941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.879526941\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.15T + 3T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 7 | \( 1 - 2.97T + 7T^{2} \) |
| 11 | \( 1 - 5.83T + 11T^{2} \) |
| 13 | \( 1 - 0.490T + 13T^{2} \) |
| 17 | \( 1 - 1.66T + 17T^{2} \) |
| 19 | \( 1 + 5.00T + 19T^{2} \) |
| 23 | \( 1 - 4.18T + 23T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 1.74T + 41T^{2} \) |
| 43 | \( 1 + 3.05T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 + 8.49T + 53T^{2} \) |
| 59 | \( 1 + 8.45T + 59T^{2} \) |
| 61 | \( 1 + 0.659T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 4.34T + 79T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−8.333894143703245202750937385655, −7.81923660149061108617641486252, −7.16044123659478703022524387164, −6.57102763410976132994862902702, −5.03173929534168591931094158882, −4.30227150330942076297436287463, −3.71629773968227602697893003581, −3.39595348114611069180372680134, −2.09767576120442753424196226431, −1.34247099282063778858347077430,
1.34247099282063778858347077430, 2.09767576120442753424196226431, 3.39595348114611069180372680134, 3.71629773968227602697893003581, 4.30227150330942076297436287463, 5.03173929534168591931094158882, 6.57102763410976132994862902702, 7.16044123659478703022524387164, 7.81923660149061108617641486252, 8.333894143703245202750937385655
