| L(s) = 1 | − 0.163·2-s + 3.13·3-s − 1.97·4-s − 0.989·5-s − 0.511·6-s − 3.57·7-s + 0.647·8-s + 6.85·9-s + 0.161·10-s − 1.26·11-s − 6.19·12-s − 2.82·13-s + 0.583·14-s − 3.10·15-s + 3.84·16-s − 3.72·17-s − 1.11·18-s − 19-s + 1.95·20-s − 11.2·21-s + 0.206·22-s + 8.50·23-s + 2.03·24-s − 4.02·25-s + 0.460·26-s + 12.1·27-s + 7.06·28-s + ⋯ |
| L(s) = 1 | − 0.115·2-s + 1.81·3-s − 0.986·4-s − 0.442·5-s − 0.208·6-s − 1.35·7-s + 0.229·8-s + 2.28·9-s + 0.0510·10-s − 0.381·11-s − 1.78·12-s − 0.783·13-s + 0.155·14-s − 0.801·15-s + 0.960·16-s − 0.902·17-s − 0.263·18-s − 0.229·19-s + 0.436·20-s − 2.45·21-s + 0.0440·22-s + 1.77·23-s + 0.415·24-s − 0.804·25-s + 0.0903·26-s + 2.32·27-s + 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.814896902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.814896902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 + 0.163T + 2T^{2} \) |
| 3 | \( 1 - 3.13T + 3T^{2} \) |
| 5 | \( 1 + 0.989T + 5T^{2} \) |
| 7 | \( 1 + 3.57T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 3.72T + 17T^{2} \) |
| 23 | \( 1 - 8.50T + 23T^{2} \) |
| 29 | \( 1 + 6.13T + 29T^{2} \) |
| 31 | \( 1 - 9.25T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 2.22T + 43T^{2} \) |
| 47 | \( 1 - 5.86T + 47T^{2} \) |
| 53 | \( 1 + 6.07T + 53T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 + 6.16T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 8.02T + 71T^{2} \) |
| 73 | \( 1 - 5.51T + 73T^{2} \) |
| 79 | \( 1 + 9.26T + 79T^{2} \) |
| 83 | \( 1 + 8.32T + 83T^{2} \) |
| 89 | \( 1 + 2.38T + 89T^{2} \) |
| 97 | \( 1 - 6.98T + 97T^{2} \) |
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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
−8.222594915857801522195489340911, −7.49273268064253512283445463323, −7.05344625261502316635493617218, −6.04578755416108431951480462288, −4.82397314162622737688029021718, −4.27081671290770810416179648100, −3.48623740036169016022795938911, −2.95288607065088087163897570834, −2.15813437189363671754947263318, −0.64018206023052678620283683121,
0.64018206023052678620283683121, 2.15813437189363671754947263318, 2.95288607065088087163897570834, 3.48623740036169016022795938911, 4.27081671290770810416179648100, 4.82397314162622737688029021718, 6.04578755416108431951480462288, 7.05344625261502316635493617218, 7.49273268064253512283445463323, 8.222594915857801522195489340911
