| L(s) = 1 | − 2.63·2-s − 0.151·3-s + 4.92·4-s − 0.321·5-s + 0.398·6-s − 0.565·7-s − 7.68·8-s − 2.97·9-s + 0.845·10-s − 4.63·11-s − 0.746·12-s − 0.0639·13-s + 1.48·14-s + 0.0487·15-s + 10.3·16-s + 1.37·17-s + 7.83·18-s − 19-s − 1.58·20-s + 0.0857·21-s + 12.1·22-s + 6.68·23-s + 1.16·24-s − 4.89·25-s + 0.168·26-s + 0.906·27-s − 2.78·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.0875·3-s + 2.46·4-s − 0.143·5-s + 0.162·6-s − 0.213·7-s − 2.71·8-s − 0.992·9-s + 0.267·10-s − 1.39·11-s − 0.215·12-s − 0.0177·13-s + 0.397·14-s + 0.0125·15-s + 2.59·16-s + 0.333·17-s + 1.84·18-s − 0.229·19-s − 0.353·20-s + 0.0187·21-s + 2.60·22-s + 1.39·23-s + 0.237·24-s − 0.979·25-s + 0.0329·26-s + 0.174·27-s − 0.525·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 + T \) |
| 317 | \( 1 + T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 + 0.151T + 3T^{2} \) |
| 5 | \( 1 + 0.321T + 5T^{2} \) |
| 7 | \( 1 + 0.565T + 7T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 13 | \( 1 + 0.0639T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 + 5.91T + 31T^{2} \) |
| 37 | \( 1 + 0.0360T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 - 2.40T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 3.00T + 67T^{2} \) |
| 71 | \( 1 - 6.28T + 71T^{2} \) |
| 73 | \( 1 - 6.93T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 4.10T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.901022043931641618878811941133, −7.29785586746218281913554776270, −6.64302402910344370584664330404, −5.69336109578661664351347889703, −5.24940301211395402269640819582, −3.70447217573388169468914501576, −2.70930849962335153045637329054, −2.27916759513343813725133031137, −0.904966884851754726216130149272, 0,
0.904966884851754726216130149272, 2.27916759513343813725133031137, 2.70930849962335153045637329054, 3.70447217573388169468914501576, 5.24940301211395402269640819582, 5.69336109578661664351347889703, 6.64302402910344370584664330404, 7.29785586746218281913554776270, 7.901022043931641618878811941133
