| L(s) = 1 | + 2.49·2-s − 1.82·3-s + 4.20·4-s + 4.09·5-s − 4.53·6-s + 3.06·7-s + 5.49·8-s + 0.318·9-s + 10.1·10-s + 1.16·11-s − 7.66·12-s + 7.63·14-s − 7.45·15-s + 5.28·16-s − 6.89·17-s + 0.792·18-s + 1.71·19-s + 17.2·20-s − 5.58·21-s + 2.89·22-s + 23-s − 10.0·24-s + 11.7·25-s + 4.88·27-s + 12.8·28-s + 7.06·29-s − 18.5·30-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 1.05·3-s + 2.10·4-s + 1.83·5-s − 1.85·6-s + 1.15·7-s + 1.94·8-s + 0.106·9-s + 3.22·10-s + 0.350·11-s − 2.21·12-s + 2.04·14-s − 1.92·15-s + 1.32·16-s − 1.67·17-s + 0.186·18-s + 0.393·19-s + 3.84·20-s − 1.21·21-s + 0.616·22-s + 0.208·23-s − 2.04·24-s + 2.34·25-s + 0.940·27-s + 2.43·28-s + 1.31·29-s − 3.39·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.359369968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.359369968\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 + 1.82T + 3T^{2} \) |
| 5 | \( 1 - 4.09T + 5T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 11 | \( 1 - 1.16T + 11T^{2} \) |
| 17 | \( 1 + 6.89T + 17T^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 29 | \( 1 - 7.06T + 29T^{2} \) |
| 31 | \( 1 + 0.519T + 31T^{2} \) |
| 37 | \( 1 + 6.65T + 37T^{2} \) |
| 41 | \( 1 - 5.67T + 41T^{2} \) |
| 43 | \( 1 + 8.61T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6.66T + 53T^{2} \) |
| 59 | \( 1 + 3.86T + 59T^{2} \) |
| 61 | \( 1 + 9.41T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 - 2.33T + 71T^{2} \) |
| 73 | \( 1 + 9.78T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 3.85T + 83T^{2} \) |
| 89 | \( 1 + 3.03T + 89T^{2} \) |
| 97 | \( 1 - 4.63T + 97T^{2} \) |
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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
−8.538749342834045680307414373119, −7.08402193605135357788107044445, −6.60942933081296029472824039205, −5.93822567417438311772919725624, −5.48087920877884121370797474310, −4.81374154911748008516746762121, −4.40207117491836843252864331957, −2.92233765141318270497192567844, −2.16311756037358082210853433858, −1.35385852763629112871540751262,
1.35385852763629112871540751262, 2.16311756037358082210853433858, 2.92233765141318270497192567844, 4.40207117491836843252864331957, 4.81374154911748008516746762121, 5.48087920877884121370797474310, 5.93822567417438311772919725624, 6.60942933081296029472824039205, 7.08402193605135357788107044445, 8.538749342834045680307414373119
