L(s) = 1 | − 2-s + 1.04·3-s + 4-s − 2.99·5-s − 1.04·6-s + 0.197·7-s − 8-s − 1.90·9-s + 2.99·10-s + 3.70·11-s + 1.04·12-s + 5.36·13-s − 0.197·14-s − 3.13·15-s + 16-s + 0.593·17-s + 1.90·18-s − 2.42·19-s − 2.99·20-s + 0.207·21-s − 3.70·22-s + 1.31·23-s − 1.04·24-s + 3.94·25-s − 5.36·26-s − 5.13·27-s + 0.197·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.604·3-s + 0.5·4-s − 1.33·5-s − 0.427·6-s + 0.0747·7-s − 0.353·8-s − 0.634·9-s + 0.945·10-s + 1.11·11-s + 0.302·12-s + 1.48·13-s − 0.0528·14-s − 0.808·15-s + 0.250·16-s + 0.143·17-s + 0.448·18-s − 0.557·19-s − 0.668·20-s + 0.0451·21-s − 0.789·22-s + 0.273·23-s − 0.213·24-s + 0.788·25-s − 1.05·26-s − 0.988·27-s + 0.0373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.242324770\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.242324770\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 - 1.04T + 3T^{2} \) |
| 5 | \( 1 + 2.99T + 5T^{2} \) |
| 7 | \( 1 - 0.197T + 7T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 - 5.36T + 13T^{2} \) |
| 17 | \( 1 - 0.593T + 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 + 8.88T + 31T^{2} \) |
| 37 | \( 1 - 2.20T + 37T^{2} \) |
| 41 | \( 1 - 6.23T + 41T^{2} \) |
| 43 | \( 1 - 3.09T + 43T^{2} \) |
| 47 | \( 1 - 5.28T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 + 8.67T + 59T^{2} \) |
| 61 | \( 1 + 4.72T + 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 - 6.22T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 2.35T + 83T^{2} \) |
| 89 | \( 1 + 0.557T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.580535436724197635347461706131, −7.84714229017015100878650354633, −7.29325762232533480842028134002, −6.37345825946327206446096396352, −5.72081080761328664907893446814, −4.30460859659446789991969782071, −3.71759520804400844792305354049, −3.08610769903557919410496489841, −1.83570140827459685201203142864, −0.69729094257629431869293034867,
0.69729094257629431869293034867, 1.83570140827459685201203142864, 3.08610769903557919410496489841, 3.71759520804400844792305354049, 4.30460859659446789991969782071, 5.72081080761328664907893446814, 6.37345825946327206446096396352, 7.29325762232533480842028134002, 7.84714229017015100878650354633, 8.580535436724197635347461706131