L(s) = 1 | − 2-s − 1.70·3-s + 4-s − 5-s + 1.70·6-s − 1.19·7-s − 8-s − 0.0930·9-s + 10-s + 5.50·11-s − 1.70·12-s + 5.99·13-s + 1.19·14-s + 1.70·15-s + 16-s + 4.84·17-s + 0.0930·18-s + 3.12·19-s − 20-s + 2.03·21-s − 5.50·22-s + 9.08·23-s + 1.70·24-s + 25-s − 5.99·26-s + 5.27·27-s − 1.19·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.984·3-s + 0.5·4-s − 0.447·5-s + 0.696·6-s − 0.452·7-s − 0.353·8-s − 0.0310·9-s + 0.316·10-s + 1.65·11-s − 0.492·12-s + 1.66·13-s + 0.319·14-s + 0.440·15-s + 0.250·16-s + 1.17·17-s + 0.0219·18-s + 0.715·19-s − 0.223·20-s + 0.445·21-s − 1.17·22-s + 1.89·23-s + 0.348·24-s + 0.200·25-s − 1.17·26-s + 1.01·27-s − 0.226·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.092169039\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.092169039\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 - 5.50T + 11T^{2} \) |
| 13 | \( 1 - 5.99T + 13T^{2} \) |
| 17 | \( 1 - 4.84T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 - 9.08T + 23T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 - 7.24T + 37T^{2} \) |
| 41 | \( 1 - 1.08T + 41T^{2} \) |
| 43 | \( 1 + 4.46T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 + 4.39T + 53T^{2} \) |
| 59 | \( 1 + 8.40T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 16.9T + 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 + 9.63T + 83T^{2} \) |
| 89 | \( 1 + 0.103T + 89T^{2} \) |
| 97 | \( 1 - 9.42T + 97T^{2} \) |
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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
−8.476478916503290469501365574155, −7.78753038769177649656813364681, −6.68216932844235427634685243822, −6.50239232368827882329182968997, −5.68616930813441591021285751606, −4.83136122835737907353115494179, −3.57380087598764484450776624288, −3.21831612779198560115625702950, −1.34841685098592053140924309279, −0.828495212314990659727092665910,
0.828495212314990659727092665910, 1.34841685098592053140924309279, 3.21831612779198560115625702950, 3.57380087598764484450776624288, 4.83136122835737907353115494179, 5.68616930813441591021285751606, 6.50239232368827882329182968997, 6.68216932844235427634685243822, 7.78753038769177649656813364681, 8.476478916503290469501365574155