| L(s) = 1 | − 1.33·2-s − 3-s − 0.215·4-s + 1.69·5-s + 1.33·6-s + 7-s + 2.95·8-s + 9-s − 2.26·10-s + 3.75·11-s + 0.215·12-s + 5.32·13-s − 1.33·14-s − 1.69·15-s − 3.52·16-s − 2.49·17-s − 1.33·18-s + 7.37·19-s − 0.366·20-s − 21-s − 5.01·22-s − 5.92·23-s − 2.95·24-s − 2.11·25-s − 7.10·26-s − 27-s − 0.215·28-s + ⋯ |
| L(s) = 1 | − 0.944·2-s − 0.577·3-s − 0.107·4-s + 0.759·5-s + 0.545·6-s + 0.377·7-s + 1.04·8-s + 0.333·9-s − 0.716·10-s + 1.13·11-s + 0.0623·12-s + 1.47·13-s − 0.356·14-s − 0.438·15-s − 0.880·16-s − 0.605·17-s − 0.314·18-s + 1.69·19-s − 0.0819·20-s − 0.218·21-s − 1.06·22-s − 1.23·23-s − 0.604·24-s − 0.423·25-s − 1.39·26-s − 0.192·27-s − 0.0408·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.411840180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.411840180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 1.33T + 2T^{2} \) |
| 5 | \( 1 - 1.69T + 5T^{2} \) |
| 11 | \( 1 - 3.75T + 11T^{2} \) |
| 13 | \( 1 - 5.32T + 13T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 23 | \( 1 + 5.92T + 23T^{2} \) |
| 29 | \( 1 + 5.24T + 29T^{2} \) |
| 31 | \( 1 - 3.55T + 31T^{2} \) |
| 37 | \( 1 + 3.75T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 - 4.38T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 4.93T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 - 3.19T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 6.28T + 83T^{2} \) |
| 89 | \( 1 - 8.53T + 89T^{2} \) |
| 97 | \( 1 + 1.20T + 97T^{2} \) |
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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
−7.87666347816197108181554799769, −7.22897060288774239458852174790, −6.49349809894710233564157267394, −5.72300273335607453362465951243, −5.32160405865455393331546725178, −4.03992991103117039670706794596, −3.88684376191004275168087271696, −2.22771397405342200953929499066, −1.41358828590344027188987096823, −0.809613890106367581029493611147,
0.809613890106367581029493611147, 1.41358828590344027188987096823, 2.22771397405342200953929499066, 3.88684376191004275168087271696, 4.03992991103117039670706794596, 5.32160405865455393331546725178, 5.72300273335607453362465951243, 6.49349809894710233564157267394, 7.22897060288774239458852174790, 7.87666347816197108181554799769
