L(s) = 1 | − 2.67·2-s + 1.79·3-s + 5.17·4-s + 3.40·5-s − 4.80·6-s + 4.24·7-s − 8.51·8-s + 0.221·9-s − 9.12·10-s + 2.35·11-s + 9.29·12-s − 0.771·13-s − 11.3·14-s + 6.11·15-s + 12.4·16-s − 0.943·17-s − 0.593·18-s + 0.177·19-s + 17.6·20-s + 7.61·21-s − 6.30·22-s + 4.24·23-s − 15.2·24-s + 6.60·25-s + 2.06·26-s − 4.98·27-s + 21.9·28-s + ⋯ |
L(s) = 1 | − 1.89·2-s + 1.03·3-s + 2.58·4-s + 1.52·5-s − 1.96·6-s + 1.60·7-s − 3.01·8-s + 0.0738·9-s − 2.88·10-s + 0.709·11-s + 2.68·12-s − 0.214·13-s − 3.03·14-s + 1.57·15-s + 3.11·16-s − 0.228·17-s − 0.139·18-s + 0.0407·19-s + 3.94·20-s + 1.66·21-s − 1.34·22-s + 0.884·23-s − 3.11·24-s + 1.32·25-s + 0.405·26-s − 0.959·27-s + 4.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.039344969\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.039344969\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4019 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 13 | \( 1 + 0.771T + 13T^{2} \) |
| 17 | \( 1 + 0.943T + 17T^{2} \) |
| 19 | \( 1 - 0.177T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 - 5.04T + 41T^{2} \) |
| 43 | \( 1 - 2.14T + 43T^{2} \) |
| 47 | \( 1 + 2.93T + 47T^{2} \) |
| 53 | \( 1 + 5.59T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 4.00T + 61T^{2} \) |
| 67 | \( 1 + 5.93T + 67T^{2} \) |
| 71 | \( 1 - 0.981T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 8.26T + 79T^{2} \) |
| 83 | \( 1 - 2.07T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 6.37T + 97T^{2} \) |
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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
−8.638967760001456192412182791723, −7.947030866660494474261040089367, −7.40040705748434605856484862832, −6.52523651156293147805873076479, −5.77554136527861438122780916240, −4.83758298513951464589382340054, −3.30395715805299099392377643007, −2.26145611917879545628574888278, −1.91311725638600173163273604964, −1.12275727886688914394181968038,
1.12275727886688914394181968038, 1.91311725638600173163273604964, 2.26145611917879545628574888278, 3.30395715805299099392377643007, 4.83758298513951464589382340054, 5.77554136527861438122780916240, 6.52523651156293147805873076479, 7.40040705748434605856484862832, 7.947030866660494474261040089367, 8.638967760001456192412182791723