| L(s) = 1 | − 2.40·2-s − 0.556·3-s + 3.78·4-s + 1.68·5-s + 1.33·6-s + 2.28·7-s − 4.30·8-s − 2.69·9-s − 4.04·10-s + 2.91·11-s − 2.10·12-s + 6.13·13-s − 5.49·14-s − 0.936·15-s + 2.78·16-s − 7.57·17-s + 6.47·18-s + 8.35·19-s + 6.37·20-s − 1.27·21-s − 7.02·22-s − 4.30·23-s + 2.39·24-s − 2.16·25-s − 14.7·26-s + 3.16·27-s + 8.65·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.321·3-s + 1.89·4-s + 0.752·5-s + 0.546·6-s + 0.863·7-s − 1.52·8-s − 0.896·9-s − 1.28·10-s + 0.879·11-s − 0.608·12-s + 1.70·13-s − 1.46·14-s − 0.241·15-s + 0.695·16-s − 1.83·17-s + 1.52·18-s + 1.91·19-s + 1.42·20-s − 0.277·21-s − 1.49·22-s − 0.897·23-s + 0.489·24-s − 0.433·25-s − 2.89·26-s + 0.609·27-s + 1.63·28-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
| \(L(1)\) |
\(\approx\) |
\(1.023532512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.023532512\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 + 0.556T + 3T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 13 | \( 1 - 6.13T + 13T^{2} \) |
| 17 | \( 1 + 7.57T + 17T^{2} \) |
| 19 | \( 1 - 8.35T + 19T^{2} \) |
| 23 | \( 1 + 4.30T + 23T^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 + 0.874T + 41T^{2} \) |
| 43 | \( 1 - 0.103T + 43T^{2} \) |
| 47 | \( 1 - 0.979T + 47T^{2} \) |
| 53 | \( 1 - 2.46T + 53T^{2} \) |
| 59 | \( 1 - 7.94T + 59T^{2} \) |
| 61 | \( 1 + 0.119T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 + 5.80T + 73T^{2} \) |
| 79 | \( 1 - 3.24T + 79T^{2} \) |
| 83 | \( 1 - 1.20T + 83T^{2} \) |
| 89 | \( 1 - 3.53T + 89T^{2} \) |
| 97 | \( 1 + 4.69T + 97T^{2} \) |
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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
−8.523117479232730006187967968741, −8.093310917601124506543352102263, −7.07816689732298899965873723455, −6.30368309770440555385840241992, −5.91948952110120537778854387268, −4.84271243603292324947939514136, −3.66834695092917128513027539806, −2.41521628738579633170904213911, −1.61699274608192906357961284719, −0.813804201774241198438166792287,
0.813804201774241198438166792287, 1.61699274608192906357961284719, 2.41521628738579633170904213911, 3.66834695092917128513027539806, 4.84271243603292324947939514136, 5.91948952110120537778854387268, 6.30368309770440555385840241992, 7.07816689732298899965873723455, 8.093310917601124506543352102263, 8.523117479232730006187967968741
