L(s) = 1 | − 0.103·2-s + 3-s − 1.98·4-s − 2.25·5-s − 0.103·6-s + 4.11·7-s + 0.412·8-s + 9-s + 0.233·10-s + 3.41·11-s − 1.98·12-s + 3.88·13-s − 0.424·14-s − 2.25·15-s + 3.93·16-s + 17-s − 0.103·18-s − 2.55·19-s + 4.48·20-s + 4.11·21-s − 0.353·22-s + 2.82·23-s + 0.412·24-s + 0.0923·25-s − 0.401·26-s + 27-s − 8.17·28-s + ⋯ |
L(s) = 1 | − 0.0730·2-s + 0.577·3-s − 0.994·4-s − 1.00·5-s − 0.0421·6-s + 1.55·7-s + 0.145·8-s + 0.333·9-s + 0.0737·10-s + 1.02·11-s − 0.574·12-s + 1.07·13-s − 0.113·14-s − 0.582·15-s + 0.984·16-s + 0.242·17-s − 0.0243·18-s − 0.586·19-s + 1.00·20-s + 0.897·21-s − 0.0752·22-s + 0.589·23-s + 0.0841·24-s + 0.0184·25-s − 0.0786·26-s + 0.192·27-s − 1.54·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.110434847\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.110434847\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 0.103T + 2T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 - 4.11T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 - 3.88T + 13T^{2} \) |
| 19 | \( 1 + 2.55T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 9.03T + 29T^{2} \) |
| 31 | \( 1 - 2.07T + 31T^{2} \) |
| 37 | \( 1 - 0.222T + 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 0.936T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 - 8.03T + 59T^{2} \) |
| 61 | \( 1 - 1.23T + 61T^{2} \) |
| 67 | \( 1 + 5.75T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 9.54T + 73T^{2} \) |
| 83 | \( 1 - 7.96T + 83T^{2} \) |
| 89 | \( 1 - 0.331T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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
−8.416573934092287313410647397669, −8.149320257112475149941774597414, −7.18513551942727134899896724552, −6.32856217893966083930727775281, −5.11196388958246587072339423577, −4.55536077631780489526528568021, −3.91623326885732624894126512550, −3.27591555514979695880352230487, −1.71314437977392451508145501802, −0.911101847982377059564661825922,
0.911101847982377059564661825922, 1.71314437977392451508145501802, 3.27591555514979695880352230487, 3.91623326885732624894126512550, 4.55536077631780489526528568021, 5.11196388958246587072339423577, 6.32856217893966083930727775281, 7.18513551942727134899896724552, 8.149320257112475149941774597414, 8.416573934092287313410647397669