| L(s) = 1 | − 2.60·2-s − 3.36·3-s + 4.77·4-s + 3.49·5-s + 8.75·6-s + 4.61·7-s − 7.21·8-s + 8.31·9-s − 9.09·10-s + 5.76·11-s − 16.0·12-s − 3.60·13-s − 12.0·14-s − 11.7·15-s + 9.22·16-s − 6.25·17-s − 21.6·18-s + 1.10·19-s + 16.6·20-s − 15.5·21-s − 14.9·22-s − 5.25·23-s + 24.2·24-s + 7.20·25-s + 9.38·26-s − 17.8·27-s + 22.0·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 1.94·3-s + 2.38·4-s + 1.56·5-s + 3.57·6-s + 1.74·7-s − 2.54·8-s + 2.77·9-s − 2.87·10-s + 1.73·11-s − 4.63·12-s − 0.999·13-s − 3.21·14-s − 3.03·15-s + 2.30·16-s − 1.51·17-s − 5.10·18-s + 0.252·19-s + 3.72·20-s − 3.38·21-s − 3.19·22-s − 1.09·23-s + 4.95·24-s + 1.44·25-s + 1.83·26-s − 3.44·27-s + 4.16·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 + 3.36T + 3T^{2} \) |
| 5 | \( 1 - 3.49T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 + 3.60T + 13T^{2} \) |
| 17 | \( 1 + 6.25T + 17T^{2} \) |
| 19 | \( 1 - 1.10T + 19T^{2} \) |
| 23 | \( 1 + 5.25T + 23T^{2} \) |
| 29 | \( 1 + 0.286T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 - 1.67T + 41T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 + 6.65T + 47T^{2} \) |
| 53 | \( 1 + 8.93T + 53T^{2} \) |
| 59 | \( 1 - 0.421T + 59T^{2} \) |
| 61 | \( 1 + 5.32T + 61T^{2} \) |
| 67 | \( 1 + 4.40T + 67T^{2} \) |
| 73 | \( 1 + 5.64T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 9.41T + 83T^{2} \) |
| 89 | \( 1 + 5.01T + 89T^{2} \) |
| 97 | \( 1 + 5.93T + 97T^{2} \) |
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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
−7.36426834123438665166294182734, −6.79717664058883446652989535430, −6.26766988064289462126294955957, −5.75205597498440776392650386191, −4.89260583708859000327084849204, −4.30229969471467922263473465008, −2.19617340521938685112067388430, −1.62739567095186412764030906376, −1.30998973576445864733467271147, 0,
1.30998973576445864733467271147, 1.62739567095186412764030906376, 2.19617340521938685112067388430, 4.30229969471467922263473465008, 4.89260583708859000327084849204, 5.75205597498440776392650386191, 6.26766988064289462126294955957, 6.79717664058883446652989535430, 7.36426834123438665166294182734
