| L(s) = 1 | − 1.66·2-s + 1.93·3-s + 0.767·4-s − 2.27·5-s − 3.22·6-s + 0.280·7-s + 2.05·8-s + 0.748·9-s + 3.79·10-s − 0.838·11-s + 1.48·12-s − 6.44·13-s − 0.466·14-s − 4.41·15-s − 4.94·16-s − 5.62·17-s − 1.24·18-s + 7.99·19-s − 1.74·20-s + 0.542·21-s + 1.39·22-s + 1.07·23-s + 3.96·24-s + 0.194·25-s + 10.7·26-s − 4.35·27-s + 0.215·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 1.11·3-s + 0.383·4-s − 1.01·5-s − 1.31·6-s + 0.105·7-s + 0.724·8-s + 0.249·9-s + 1.19·10-s − 0.252·11-s + 0.428·12-s − 1.78·13-s − 0.124·14-s − 1.13·15-s − 1.23·16-s − 1.36·17-s − 0.293·18-s + 1.83·19-s − 0.391·20-s + 0.118·21-s + 0.297·22-s + 0.224·23-s + 0.810·24-s + 0.0389·25-s + 2.10·26-s − 0.839·27-s + 0.0406·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6939528893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6939528893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 1.66T + 2T^{2} \) |
| 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 7 | \( 1 - 0.280T + 7T^{2} \) |
| 11 | \( 1 + 0.838T + 11T^{2} \) |
| 13 | \( 1 + 6.44T + 13T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 - 7.99T + 19T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 - 7.63T + 29T^{2} \) |
| 31 | \( 1 + 6.25T + 31T^{2} \) |
| 41 | \( 1 + 0.298T + 41T^{2} \) |
| 43 | \( 1 + 2.25T + 43T^{2} \) |
| 47 | \( 1 - 4.42T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 3.13T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 7.64T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 0.322T + 97T^{2} \) |
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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
−8.270295789252814928487158022776, −8.018031841764862276334941355106, −7.29739514855697713463618194164, −6.89010463953015863860489763041, −5.22484837018471242592931858969, −4.61503201683547001381301950395, −3.66736882752272456524697530594, −2.75306565500620471431554966588, −1.98978762454632040193923157638, −0.51855555651288072798143041507,
0.51855555651288072798143041507, 1.98978762454632040193923157638, 2.75306565500620471431554966588, 3.66736882752272456524697530594, 4.61503201683547001381301950395, 5.22484837018471242592931858969, 6.89010463953015863860489763041, 7.29739514855697713463618194164, 8.018031841764862276334941355106, 8.270295789252814928487158022776
