| L(s) = 1 | + 0.0750·2-s + 3.37·3-s − 1.99·4-s − 5-s + 0.253·6-s + 1.31·7-s − 0.299·8-s + 8.38·9-s − 0.0750·10-s − 0.660·11-s − 6.72·12-s + 3.48·13-s + 0.0987·14-s − 3.37·15-s + 3.96·16-s + 3.95·17-s + 0.629·18-s − 3.86·19-s + 1.99·20-s + 4.43·21-s − 0.0496·22-s − 2.37·23-s − 1.01·24-s + 25-s + 0.261·26-s + 18.1·27-s − 2.62·28-s + ⋯ |
| L(s) = 1 | + 0.0530·2-s + 1.94·3-s − 0.997·4-s − 0.447·5-s + 0.103·6-s + 0.497·7-s − 0.106·8-s + 2.79·9-s − 0.0237·10-s − 0.199·11-s − 1.94·12-s + 0.967·13-s + 0.0263·14-s − 0.871·15-s + 0.991·16-s + 0.959·17-s + 0.148·18-s − 0.887·19-s + 0.445·20-s + 0.968·21-s − 0.0105·22-s − 0.495·23-s − 0.206·24-s + 0.200·25-s + 0.0513·26-s + 3.49·27-s − 0.495·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.095863690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.095863690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0750T + 2T^{2} \) |
| 3 | \( 1 - 3.37T + 3T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 + 0.660T + 11T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 17 | \( 1 - 3.95T + 17T^{2} \) |
| 19 | \( 1 + 3.86T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 - 5.63T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 0.449T + 43T^{2} \) |
| 47 | \( 1 + 8.97T + 47T^{2} \) |
| 53 | \( 1 - 9.21T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 3.38T + 61T^{2} \) |
| 67 | \( 1 + 5.41T + 67T^{2} \) |
| 71 | \( 1 - 0.493T + 71T^{2} \) |
| 73 | \( 1 + 1.48T + 73T^{2} \) |
| 79 | \( 1 + 1.31T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 7.11T + 97T^{2} \) |
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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
−8.994647010446260367143532205766, −8.312082312386536196236591162173, −7.980951307061737353009595462048, −7.26538837542384917230957431834, −5.92936469634539624522199751453, −4.73705848472355281729589582564, −3.92173432983918329383515625001, −3.54523681980726886359664616552, −2.36574731891971097694762566903, −1.18920239847552767803075034036,
1.18920239847552767803075034036, 2.36574731891971097694762566903, 3.54523681980726886359664616552, 3.92173432983918329383515625001, 4.73705848472355281729589582564, 5.92936469634539624522199751453, 7.26538837542384917230957431834, 7.980951307061737353009595462048, 8.312082312386536196236591162173, 8.994647010446260367143532205766
