| L(s) = 1 | − 2.39·2-s + 3.13·3-s + 3.73·4-s + 0.529·5-s − 7.50·6-s − 1.46·7-s − 4.16·8-s + 6.80·9-s − 1.26·10-s + 4.42·11-s + 11.7·12-s + 6.46·13-s + 3.49·14-s + 1.65·15-s + 2.50·16-s + 1.71·17-s − 16.3·18-s − 1.59·19-s + 1.98·20-s − 4.57·21-s − 10.6·22-s + 5.85·23-s − 13.0·24-s − 4.71·25-s − 15.4·26-s + 11.9·27-s − 5.46·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.80·3-s + 1.86·4-s + 0.237·5-s − 3.06·6-s − 0.551·7-s − 1.47·8-s + 2.26·9-s − 0.401·10-s + 1.33·11-s + 3.38·12-s + 1.79·13-s + 0.935·14-s + 0.428·15-s + 0.626·16-s + 0.415·17-s − 3.84·18-s − 0.366·19-s + 0.443·20-s − 0.997·21-s − 2.26·22-s + 1.22·23-s − 2.66·24-s − 0.943·25-s − 3.03·26-s + 2.29·27-s − 1.03·28-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
| \(L(1)\) |
\(\approx\) |
\(2.148183279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148183279\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.39T + 2T^{2} \) |
| 3 | \( 1 - 3.13T + 3T^{2} \) |
| 5 | \( 1 - 0.529T + 5T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 - 6.46T + 13T^{2} \) |
| 17 | \( 1 - 1.71T + 17T^{2} \) |
| 19 | \( 1 + 1.59T + 19T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 + 6.42T + 29T^{2} \) |
| 31 | \( 1 + 2.54T + 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 2.30T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 + 0.795T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 9.51T + 71T^{2} \) |
| 73 | \( 1 + 0.209T + 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 + 6.61T + 83T^{2} \) |
| 89 | \( 1 + 9.99T + 89T^{2} \) |
| 97 | \( 1 + 5.09T + 97T^{2} \) |
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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
−8.762703037781034646007904327033, −7.935170162479921967883814493602, −7.40300517009390570602018121262, −6.59390711495822702129364093554, −6.00116592414937126363215790898, −4.14173441248260249125118646232, −3.57011113657180734680008089100, −2.70712890275420405613065066957, −1.69489226377158933128111539289, −1.12356542358511636904238697348,
1.12356542358511636904238697348, 1.69489226377158933128111539289, 2.70712890275420405613065066957, 3.57011113657180734680008089100, 4.14173441248260249125118646232, 6.00116592414937126363215790898, 6.59390711495822702129364093554, 7.40300517009390570602018121262, 7.935170162479921967883814493602, 8.762703037781034646007904327033
