| L(s) = 1 | − 2.25·2-s + 2.53·3-s + 3.07·4-s − 2.69·5-s − 5.71·6-s − 2.07·7-s − 2.42·8-s + 3.44·9-s + 6.07·10-s + 11-s + 7.80·12-s + 1.22·13-s + 4.66·14-s − 6.84·15-s − 0.695·16-s + 17-s − 7.75·18-s + 2.33·19-s − 8.29·20-s − 5.26·21-s − 2.25·22-s + 3.99·23-s − 6.14·24-s + 2.27·25-s − 2.75·26-s + 1.12·27-s − 6.37·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 1.46·3-s + 1.53·4-s − 1.20·5-s − 2.33·6-s − 0.783·7-s − 0.855·8-s + 1.14·9-s + 1.92·10-s + 0.301·11-s + 2.25·12-s + 0.338·13-s + 1.24·14-s − 1.76·15-s − 0.173·16-s + 0.242·17-s − 1.82·18-s + 0.536·19-s − 1.85·20-s − 1.14·21-s − 0.480·22-s + 0.833·23-s − 1.25·24-s + 0.455·25-s − 0.539·26-s + 0.217·27-s − 1.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9909346685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9909346685\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 19 | \( 1 - 2.33T + 19T^{2} \) |
| 23 | \( 1 - 3.99T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 - 5.27T + 31T^{2} \) |
| 37 | \( 1 + 3.32T + 37T^{2} \) |
| 41 | \( 1 + 9.84T + 41T^{2} \) |
| 47 | \( 1 - 9.90T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 5.02T + 59T^{2} \) |
| 61 | \( 1 + 5.30T + 61T^{2} \) |
| 67 | \( 1 + 3.52T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 5.09T + 73T^{2} \) |
| 79 | \( 1 - 6.37T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 8.54T + 97T^{2} \) |
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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
−7.85362298298185860213627695540, −7.56232209650770526096393709215, −6.96849779871833401781050110154, −6.20126108775820978053254494371, −4.85173219503172367947004346671, −3.81439585079894488525977244959, −3.35928687367402130943317205586, −2.62342033574637723444652292497, −1.61196252391017024965605238370, −0.59238626152387339527755444847,
0.59238626152387339527755444847, 1.61196252391017024965605238370, 2.62342033574637723444652292497, 3.35928687367402130943317205586, 3.81439585079894488525977244959, 4.85173219503172367947004346671, 6.20126108775820978053254494371, 6.96849779871833401781050110154, 7.56232209650770526096393709215, 7.85362298298185860213627695540
