L(s) = 1 | − 0.381·2-s + 3-s − 1.85·4-s − 3·5-s − 0.381·6-s − 4.61·7-s + 1.47·8-s + 9-s + 1.14·10-s − 5.47·11-s − 1.85·12-s + 6.70·13-s + 1.76·14-s − 3·15-s + 3.14·16-s − 0.381·17-s − 0.381·18-s + 0.854·19-s + 5.56·20-s − 4.61·21-s + 2.09·22-s − 3.61·23-s + 1.47·24-s + 4·25-s − 2.56·26-s + 27-s + 8.56·28-s + ⋯ |
L(s) = 1 | − 0.270·2-s + 0.577·3-s − 0.927·4-s − 1.34·5-s − 0.155·6-s − 1.74·7-s + 0.520·8-s + 0.333·9-s + 0.362·10-s − 1.64·11-s − 0.535·12-s + 1.86·13-s + 0.471·14-s − 0.774·15-s + 0.786·16-s − 0.0926·17-s − 0.0900·18-s + 0.195·19-s + 1.24·20-s − 1.00·21-s + 0.445·22-s − 0.754·23-s + 0.300·24-s + 0.800·25-s − 0.502·26-s + 0.192·27-s + 1.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 17 | \( 1 + 0.381T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 + 6.09T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 + 0.708T + 43T^{2} \) |
| 47 | \( 1 - 1.85T + 47T^{2} \) |
| 53 | \( 1 + 3.47T + 53T^{2} \) |
| 61 | \( 1 - 0.909T + 61T^{2} \) |
| 67 | \( 1 - 0.236T + 67T^{2} \) |
| 71 | \( 1 + 0.236T + 71T^{2} \) |
| 73 | \( 1 + 3.09T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 7.85T + 83T^{2} \) |
| 89 | \( 1 + 5.09T + 89T^{2} \) |
| 97 | \( 1 + 0.527T + 97T^{2} \) |
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
−12.60558887717726461285525209710, −11.00838594179521865888427482558, −10.08024754983485389161660819524, −9.025016345262261771921285046499, −8.210294800185043085469038168256, −7.34020609643612936293978424441, −5.74102158832280319397128200092, −3.97306476184463346821790170712, −3.31999780428615414065109765400, 0,
3.31999780428615414065109765400, 3.97306476184463346821790170712, 5.74102158832280319397128200092, 7.34020609643612936293978424441, 8.210294800185043085469038168256, 9.025016345262261771921285046499, 10.08024754983485389161660819524, 11.00838594179521865888427482558, 12.60558887717726461285525209710