| L(s) = 1 | − 0.255·2-s − 0.387·3-s − 1.93·4-s + 2.57·5-s + 0.0991·6-s − 0.612·7-s + 1.00·8-s − 2.84·9-s − 0.659·10-s − 11-s + 0.750·12-s − 3.42·13-s + 0.156·14-s − 15-s + 3.61·16-s + 1.40·17-s + 0.728·18-s − 0.203·19-s − 4.98·20-s + 0.237·21-s + 0.255·22-s − 7.26·23-s − 0.390·24-s + 1.64·25-s + 0.874·26-s + 2.26·27-s + 1.18·28-s + ⋯ |
| L(s) = 1 | − 0.180·2-s − 0.223·3-s − 0.967·4-s + 1.15·5-s + 0.0404·6-s − 0.231·7-s + 0.355·8-s − 0.949·9-s − 0.208·10-s − 0.301·11-s + 0.216·12-s − 0.948·13-s + 0.0418·14-s − 0.258·15-s + 0.902·16-s + 0.341·17-s + 0.171·18-s − 0.0467·19-s − 1.11·20-s + 0.0518·21-s + 0.0545·22-s − 1.51·23-s − 0.0796·24-s + 0.329·25-s + 0.171·26-s + 0.436·27-s + 0.223·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.255T + 2T^{2} \) |
| 3 | \( 1 + 0.387T + 3T^{2} \) |
| 5 | \( 1 - 2.57T + 5T^{2} \) |
| 7 | \( 1 + 0.612T + 7T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 + 0.203T + 19T^{2} \) |
| 23 | \( 1 + 7.26T + 23T^{2} \) |
| 29 | \( 1 + 2.39T + 29T^{2} \) |
| 31 | \( 1 - 0.0531T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 5.84T + 43T^{2} \) |
| 47 | \( 1 - 5.88T + 47T^{2} \) |
| 53 | \( 1 + 9.59T + 53T^{2} \) |
| 59 | \( 1 + 6.40T + 59T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 - 6.32T + 71T^{2} \) |
| 73 | \( 1 + 3.28T + 73T^{2} \) |
| 79 | \( 1 - 6.61T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 - 6.95T + 97T^{2} \) |
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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
−9.948186238402714139360541025899, −9.373671556730208233960653827835, −8.485091436537824737625786935528, −7.60112957037106491532293006890, −6.21990955294778653755394421123, −5.54528460244094820338802018239, −4.75639753735756710041031284030, −3.32227902790717779345396195603, −1.97485106343329027577602507079, 0,
1.97485106343329027577602507079, 3.32227902790717779345396195603, 4.75639753735756710041031284030, 5.54528460244094820338802018239, 6.21990955294778653755394421123, 7.60112957037106491532293006890, 8.485091436537824737625786935528, 9.373671556730208233960653827835, 9.948186238402714139360541025899
