| L(s) = 1 | − 0.529·2-s − 1.95·3-s − 1.71·4-s − 2.83·5-s + 1.03·6-s − 3.44·7-s + 1.97·8-s + 0.805·9-s + 1.50·10-s + 0.253·11-s + 3.35·12-s + 4.52·13-s + 1.82·14-s + 5.53·15-s + 2.39·16-s − 1.52·17-s − 0.427·18-s + 0.757·19-s + 4.87·20-s + 6.72·21-s − 0.134·22-s + 7.67·23-s − 3.84·24-s + 3.04·25-s − 2.39·26-s + 4.28·27-s + 5.92·28-s + ⋯ |
| L(s) = 1 | − 0.374·2-s − 1.12·3-s − 0.859·4-s − 1.26·5-s + 0.422·6-s − 1.30·7-s + 0.696·8-s + 0.268·9-s + 0.475·10-s + 0.0763·11-s + 0.968·12-s + 1.25·13-s + 0.488·14-s + 1.42·15-s + 0.598·16-s − 0.369·17-s − 0.100·18-s + 0.173·19-s + 1.09·20-s + 1.46·21-s − 0.0285·22-s + 1.59·23-s − 0.784·24-s + 0.608·25-s − 0.469·26-s + 0.823·27-s + 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4407532292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4407532292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + T \) |
| 113 | \( 1 - T \) |
| good | 2 | \( 1 + 0.529T + 2T^{2} \) |
| 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 + 2.83T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 0.253T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 + 1.52T + 17T^{2} \) |
| 19 | \( 1 - 0.757T + 19T^{2} \) |
| 23 | \( 1 - 7.67T + 23T^{2} \) |
| 29 | \( 1 - 8.19T + 29T^{2} \) |
| 31 | \( 1 - 4.43T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 - 7.60T + 41T^{2} \) |
| 43 | \( 1 + 3.07T + 43T^{2} \) |
| 47 | \( 1 + 8.04T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 7.01T + 59T^{2} \) |
| 61 | \( 1 + 7.64T + 61T^{2} \) |
| 67 | \( 1 + 0.242T + 67T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 - 4.92T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.003596158882037463292732048700, −7.01499116923778262237991093947, −6.43223534644385971191596257874, −5.91824509493328153486813139245, −4.80619529015159253405536571207, −4.51812035961745377201888318505, −3.47931363045849854809365619628, −3.06944821251555215065847354034, −1.06396731996504602963956709566, −0.47987028123727077418160124130,
0.47987028123727077418160124130, 1.06396731996504602963956709566, 3.06944821251555215065847354034, 3.47931363045849854809365619628, 4.51812035961745377201888318505, 4.80619529015159253405536571207, 5.91824509493328153486813139245, 6.43223534644385971191596257874, 7.01499116923778262237991093947, 8.003596158882037463292732048700
