| L(s) = 1 | − 0.762·2-s + 3-s − 1.41·4-s + 5-s − 0.762·6-s + 2.32·7-s + 2.60·8-s + 9-s − 0.762·10-s + 3.61·11-s − 1.41·12-s − 4.78·13-s − 1.76·14-s + 15-s + 0.853·16-s − 4.58·17-s − 0.762·18-s − 0.339·19-s − 1.41·20-s + 2.32·21-s − 2.75·22-s − 2.51·23-s + 2.60·24-s + 25-s + 3.64·26-s + 27-s − 3.29·28-s + ⋯ |
| L(s) = 1 | − 0.538·2-s + 0.577·3-s − 0.709·4-s + 0.447·5-s − 0.311·6-s + 0.877·7-s + 0.921·8-s + 0.333·9-s − 0.240·10-s + 1.08·11-s − 0.409·12-s − 1.32·13-s − 0.472·14-s + 0.258·15-s + 0.213·16-s − 1.11·17-s − 0.179·18-s − 0.0779·19-s − 0.317·20-s + 0.506·21-s − 0.586·22-s − 0.525·23-s + 0.531·24-s + 0.200·25-s + 0.715·26-s + 0.192·27-s − 0.622·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.762T + 2T^{2} \) |
| 7 | \( 1 - 2.32T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 4.78T + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 + 0.339T + 19T^{2} \) |
| 23 | \( 1 + 2.51T + 23T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 + 7.45T + 31T^{2} \) |
| 37 | \( 1 - 0.818T + 37T^{2} \) |
| 41 | \( 1 + 9.15T + 41T^{2} \) |
| 43 | \( 1 - 6.57T + 43T^{2} \) |
| 47 | \( 1 + 7.59T + 47T^{2} \) |
| 53 | \( 1 + 0.443T + 53T^{2} \) |
| 59 | \( 1 + 5.81T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 8.94T + 73T^{2} \) |
| 79 | \( 1 - 7.40T + 79T^{2} \) |
| 83 | \( 1 - 1.45T + 83T^{2} \) |
| 89 | \( 1 - 7.96T + 89T^{2} \) |
| 97 | \( 1 + 0.0263T + 97T^{2} \) |
| 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
−7.86238075664140490025699851502, −7.23940252647473375647411169066, −6.48658765365084543712645644010, −5.44092044200310674148228833383, −4.60036478334791195833193988269, −4.30574093899392497014131890503, −3.17755530429928573699134696837, −1.99434273227821767051531882737, −1.49932475048700211843725331016, 0,
1.49932475048700211843725331016, 1.99434273227821767051531882737, 3.17755530429928573699134696837, 4.30574093899392497014131890503, 4.60036478334791195833193988269, 5.44092044200310674148228833383, 6.48658765365084543712645644010, 7.23940252647473375647411169066, 7.86238075664140490025699851502
