| L(s) = 1 | + 1.03·2-s − 3-s − 0.921·4-s − 1.07·5-s − 1.03·6-s + 1.21·7-s − 3.03·8-s + 9-s − 1.11·10-s + 11-s + 0.921·12-s + 5.23·13-s + 1.25·14-s + 1.07·15-s − 1.30·16-s − 5.90·17-s + 1.03·18-s − 1.31·19-s + 0.988·20-s − 1.21·21-s + 1.03·22-s + 0.421·23-s + 3.03·24-s − 3.84·25-s + 5.43·26-s − 27-s − 1.11·28-s + ⋯ |
| L(s) = 1 | + 0.734·2-s − 0.577·3-s − 0.460·4-s − 0.479·5-s − 0.423·6-s + 0.457·7-s − 1.07·8-s + 0.333·9-s − 0.352·10-s + 0.301·11-s + 0.266·12-s + 1.45·13-s + 0.336·14-s + 0.276·15-s − 0.326·16-s − 1.43·17-s + 0.244·18-s − 0.302·19-s + 0.221·20-s − 0.264·21-s + 0.221·22-s + 0.0878·23-s + 0.619·24-s − 0.769·25-s + 1.06·26-s − 0.192·27-s − 0.210·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.03T + 2T^{2} \) |
| 5 | \( 1 + 1.07T + 5T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 + 5.90T + 17T^{2} \) |
| 19 | \( 1 + 1.31T + 19T^{2} \) |
| 23 | \( 1 - 0.421T + 23T^{2} \) |
| 29 | \( 1 - 9.15T + 29T^{2} \) |
| 31 | \( 1 + 0.589T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 7.76T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 67 | \( 1 + 6.66T + 67T^{2} \) |
| 71 | \( 1 + 2.90T + 71T^{2} \) |
| 73 | \( 1 + 8.81T + 73T^{2} \) |
| 79 | \( 1 + 1.13T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 6.67T + 89T^{2} \) |
| 97 | \( 1 + 6.05T + 97T^{2} \) |
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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
−8.492691738358354621013398371520, −8.311539723659163783012133418377, −6.82116124152538072351167477456, −6.33494259775990584142258352900, −5.43465410764884879646344959243, −4.55646293168627364359795240727, −4.07743428921784221243015650769, −3.12125811403305475252376225291, −1.53763001638617841916906055884, 0,
1.53763001638617841916906055884, 3.12125811403305475252376225291, 4.07743428921784221243015650769, 4.55646293168627364359795240727, 5.43465410764884879646344959243, 6.33494259775990584142258352900, 6.82116124152538072351167477456, 8.311539723659163783012133418377, 8.492691738358354621013398371520
