| L(s) = 1 | − 2.21·2-s + 3-s + 2.88·4-s + 1.51·5-s − 2.21·6-s + 0.0486·7-s − 1.96·8-s + 9-s − 3.35·10-s + 6.56·11-s + 2.88·12-s + 0.517·13-s − 0.107·14-s + 1.51·15-s − 1.43·16-s − 4.76·17-s − 2.21·18-s − 0.00792·19-s + 4.38·20-s + 0.0486·21-s − 14.5·22-s + 4.25·23-s − 1.96·24-s − 2.69·25-s − 1.14·26-s + 27-s + 0.140·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 0.577·3-s + 1.44·4-s + 0.678·5-s − 0.902·6-s + 0.0183·7-s − 0.694·8-s + 0.333·9-s − 1.06·10-s + 1.97·11-s + 0.833·12-s + 0.143·13-s − 0.0287·14-s + 0.391·15-s − 0.358·16-s − 1.15·17-s − 0.521·18-s − 0.00181·19-s + 0.980·20-s + 0.0106·21-s − 3.09·22-s + 0.886·23-s − 0.400·24-s − 0.539·25-s − 0.224·26-s + 0.192·27-s + 0.0265·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.481500148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.481500148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 5 | \( 1 - 1.51T + 5T^{2} \) |
| 7 | \( 1 - 0.0486T + 7T^{2} \) |
| 11 | \( 1 - 6.56T + 11T^{2} \) |
| 13 | \( 1 - 0.517T + 13T^{2} \) |
| 17 | \( 1 + 4.76T + 17T^{2} \) |
| 19 | \( 1 + 0.00792T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 + 7.88T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 + 8.77T + 43T^{2} \) |
| 47 | \( 1 - 1.80T + 47T^{2} \) |
| 53 | \( 1 - 5.72T + 53T^{2} \) |
| 59 | \( 1 - 0.207T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 8.59T + 67T^{2} \) |
| 71 | \( 1 - 0.245T + 71T^{2} \) |
| 73 | \( 1 - 9.82T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 6.64T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.037806271743441072809272609713, −7.24518350625726560585123537734, −6.63992479047448799248453519705, −6.25016334903479332214579519589, −5.06239455500280014607348833419, −4.09842770902792733251762627442, −3.37992564688319399506198755386, −2.05203371258239911885092002052, −1.80493551692842253308200127901, −0.76410445625170405408788654529,
0.76410445625170405408788654529, 1.80493551692842253308200127901, 2.05203371258239911885092002052, 3.37992564688319399506198755386, 4.09842770902792733251762627442, 5.06239455500280014607348833419, 6.25016334903479332214579519589, 6.63992479047448799248453519705, 7.24518350625726560585123537734, 8.037806271743441072809272609713
