| L(s) = 1 | − 2-s + 3.19·3-s + 4-s − 1.53·5-s − 3.19·6-s + 0.639·7-s − 8-s + 7.23·9-s + 1.53·10-s − 1.28·11-s + 3.19·12-s + 6.60·13-s − 0.639·14-s − 4.90·15-s + 16-s + 2.81·17-s − 7.23·18-s + 5.30·19-s − 1.53·20-s + 2.04·21-s + 1.28·22-s − 1.41·23-s − 3.19·24-s − 2.65·25-s − 6.60·26-s + 13.5·27-s + 0.639·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.84·3-s + 0.5·4-s − 0.685·5-s − 1.30·6-s + 0.241·7-s − 0.353·8-s + 2.41·9-s + 0.484·10-s − 0.388·11-s + 0.923·12-s + 1.83·13-s − 0.171·14-s − 1.26·15-s + 0.250·16-s + 0.683·17-s − 1.70·18-s + 1.21·19-s − 0.342·20-s + 0.446·21-s + 0.274·22-s − 0.294·23-s − 0.653·24-s − 0.530·25-s − 1.29·26-s + 2.61·27-s + 0.120·28-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
| \(L(1)\) |
\(\approx\) |
\(2.944056762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.944056762\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.19T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 - 0.639T + 7T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 17 | \( 1 - 2.81T + 17T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 5.59T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 + 6.45T + 37T^{2} \) |
| 41 | \( 1 - 0.109T + 41T^{2} \) |
| 43 | \( 1 - 1.49T + 43T^{2} \) |
| 47 | \( 1 - 7.09T + 47T^{2} \) |
| 53 | \( 1 + 7.70T + 53T^{2} \) |
| 59 | \( 1 + 0.321T + 59T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 - 9.17T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 - 6.26T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 7.99T + 97T^{2} \) |
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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
−8.309175810827090910405671137972, −7.935713822825422663609337156449, −7.49710116194248030371419241665, −6.57855165397223474031882873183, −5.52879170177221992328651353981, −4.26234463883408132190730120092, −3.50636931696925219966703935729, −3.08288775015771040589634089159, −1.91219226527594057619249793808, −1.09179580345674810263886564120,
1.09179580345674810263886564120, 1.91219226527594057619249793808, 3.08288775015771040589634089159, 3.50636931696925219966703935729, 4.26234463883408132190730120092, 5.52879170177221992328651353981, 6.57855165397223474031882873183, 7.49710116194248030371419241665, 7.935713822825422663609337156449, 8.309175810827090910405671137972
