| L(s) = 1 | + 0.555·2-s − 3-s − 1.69·4-s − 2.12·5-s − 0.555·6-s − 2.22·7-s − 2.05·8-s + 9-s − 1.17·10-s + 1.48·11-s + 1.69·12-s − 6.98·13-s − 1.23·14-s + 2.12·15-s + 2.24·16-s − 5.30·17-s + 0.555·18-s + 6.51·19-s + 3.59·20-s + 2.22·21-s + 0.825·22-s − 8.68·23-s + 2.05·24-s − 0.488·25-s − 3.88·26-s − 27-s + 3.75·28-s + ⋯ |
| L(s) = 1 | + 0.392·2-s − 0.577·3-s − 0.845·4-s − 0.949·5-s − 0.226·6-s − 0.839·7-s − 0.725·8-s + 0.333·9-s − 0.373·10-s + 0.448·11-s + 0.488·12-s − 1.93·13-s − 0.329·14-s + 0.548·15-s + 0.560·16-s − 1.28·17-s + 0.130·18-s + 1.49·19-s + 0.803·20-s + 0.484·21-s + 0.176·22-s − 1.81·23-s + 0.418·24-s − 0.0976·25-s − 0.760·26-s − 0.192·27-s + 0.709·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3053560997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3053560997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 677 | \( 1 - T \) |
| good | 2 | \( 1 - 0.555T + 2T^{2} \) |
| 5 | \( 1 + 2.12T + 5T^{2} \) |
| 7 | \( 1 + 2.22T + 7T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 + 6.98T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 - 6.51T + 19T^{2} \) |
| 23 | \( 1 + 8.68T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 + 6.16T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 - 8.41T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 7.45T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 0.263T + 67T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 + 1.73T + 79T^{2} \) |
| 83 | \( 1 - 1.21T + 83T^{2} \) |
| 89 | \( 1 + 1.31T + 89T^{2} \) |
| 97 | \( 1 + 2.28T + 97T^{2} \) |
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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
−9.452666562710394696827561120138, −8.344866125353894746727488024342, −7.38186629466960710604161055311, −6.91171630641409280086139583911, −5.69939748407765186715245783726, −5.16104063449272852292639102492, −4.05496565987024722004305117933, −3.76833845518315115791058970180, −2.37828720046931105089873851582, −0.33653610429892644117487122173,
0.33653610429892644117487122173, 2.37828720046931105089873851582, 3.76833845518315115791058970180, 4.05496565987024722004305117933, 5.16104063449272852292639102492, 5.69939748407765186715245783726, 6.91171630641409280086139583911, 7.38186629466960710604161055311, 8.344866125353894746727488024342, 9.452666562710394696827561120138
