| L(s) = 1 | + 2.28·2-s − 0.0897·3-s + 3.22·4-s − 5-s − 0.205·6-s − 4.22·7-s + 2.81·8-s − 2.99·9-s − 2.28·10-s − 0.289·12-s − 13-s − 9.65·14-s + 0.0897·15-s − 0.0297·16-s − 0.295·17-s − 6.84·18-s + 5.10·19-s − 3.22·20-s + 0.378·21-s + 4.20·23-s − 0.252·24-s + 25-s − 2.28·26-s + 0.537·27-s − 13.6·28-s + 1.22·29-s + 0.205·30-s + ⋯ |
| L(s) = 1 | + 1.61·2-s − 0.0517·3-s + 1.61·4-s − 0.447·5-s − 0.0837·6-s − 1.59·7-s + 0.993·8-s − 0.997·9-s − 0.723·10-s − 0.0836·12-s − 0.277·13-s − 2.58·14-s + 0.0231·15-s − 0.00743·16-s − 0.0717·17-s − 1.61·18-s + 1.17·19-s − 0.722·20-s + 0.0826·21-s + 0.876·23-s − 0.0514·24-s + 0.200·25-s − 0.448·26-s + 0.103·27-s − 2.57·28-s + 0.226·29-s + 0.0374·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.988981603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.988981603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 + 0.0897T + 3T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 17 | \( 1 + 0.295T + 17T^{2} \) |
| 19 | \( 1 - 5.10T + 19T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 29 | \( 1 - 1.22T + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 - 0.0195T + 37T^{2} \) |
| 41 | \( 1 - 4.83T + 41T^{2} \) |
| 43 | \( 1 - 2.39T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 0.166T + 53T^{2} \) |
| 59 | \( 1 - 5.42T + 59T^{2} \) |
| 61 | \( 1 - 3.36T + 61T^{2} \) |
| 67 | \( 1 + 0.529T + 67T^{2} \) |
| 71 | \( 1 - 0.0282T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 1.98T + 89T^{2} \) |
| 97 | \( 1 + 2.76T + 97T^{2} \) |
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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
−7.37561804474976590729047101749, −7.01562592970044581858076924792, −6.23270900570834417544251643270, −5.64668692318876623497569967598, −5.16464925855935021445068627745, −4.17378508999456318718620583536, −3.51780331152518765320986909042, −2.98315777433672340168562829104, −2.42580773336251946351047053393, −0.63337345240438647755420241704,
0.63337345240438647755420241704, 2.42580773336251946351047053393, 2.98315777433672340168562829104, 3.51780331152518765320986909042, 4.17378508999456318718620583536, 5.16464925855935021445068627745, 5.64668692318876623497569967598, 6.23270900570834417544251643270, 7.01562592970044581858076924792, 7.37561804474976590729047101749
