L(s) = 1 | − 1.04·2-s − 0.900·4-s + 4.37·5-s − 3.70·7-s + 3.04·8-s − 4.58·10-s − 11-s − 2.73·13-s + 3.88·14-s − 1.38·16-s + 3.57·17-s − 3.41·19-s − 3.93·20-s + 1.04·22-s + 3.58·23-s + 14.1·25-s + 2.86·26-s + 3.33·28-s + 1.76·29-s + 0.137·31-s − 4.62·32-s − 3.74·34-s − 16.2·35-s − 3.66·37-s + 3.58·38-s + 13.2·40-s − 1.23·41-s + ⋯ |
L(s) = 1 | − 0.741·2-s − 0.450·4-s + 1.95·5-s − 1.40·7-s + 1.07·8-s − 1.44·10-s − 0.301·11-s − 0.758·13-s + 1.03·14-s − 0.347·16-s + 0.866·17-s − 0.783·19-s − 0.880·20-s + 0.223·22-s + 0.747·23-s + 2.82·25-s + 0.562·26-s + 0.630·28-s + 0.327·29-s + 0.0247·31-s − 0.817·32-s − 0.642·34-s − 2.73·35-s − 0.601·37-s + 0.581·38-s + 2.10·40-s − 0.192·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.238736530\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.238736530\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 5 | \( 1 - 4.37T + 5T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 3.57T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 - 3.58T + 23T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 - 0.137T + 31T^{2} \) |
| 37 | \( 1 + 3.66T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 + 5.20T + 43T^{2} \) |
| 47 | \( 1 + 1.30T + 47T^{2} \) |
| 53 | \( 1 - 6.66T + 53T^{2} \) |
| 59 | \( 1 - 8.84T + 59T^{2} \) |
| 67 | \( 1 - 0.443T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 6.52T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 4.85T + 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.423719216825307458724837444686, −7.18810907561694180350031250448, −6.78490540856550157294034736078, −5.92946356312571691587101797155, −5.35515482675546539456826596099, −4.65948331745823528797654412904, −3.40854149866124228792995752228, −2.60685628861368968300551125203, −1.75659421136802716908502376432, −0.65922018938889865794484106532,
0.65922018938889865794484106532, 1.75659421136802716908502376432, 2.60685628861368968300551125203, 3.40854149866124228792995752228, 4.65948331745823528797654412904, 5.35515482675546539456826596099, 5.92946356312571691587101797155, 6.78490540856550157294034736078, 7.18810907561694180350031250448, 8.423719216825307458724837444686