| L(s) = 1 | − 2-s − 2.19·3-s + 4-s − 2.77·5-s + 2.19·6-s + 3.06·7-s − 8-s + 1.79·9-s + 2.77·10-s − 4.27·11-s − 2.19·12-s + 6.31·13-s − 3.06·14-s + 6.06·15-s + 16-s − 5.67·17-s − 1.79·18-s + 3.05·19-s − 2.77·20-s − 6.72·21-s + 4.27·22-s − 3.16·23-s + 2.19·24-s + 2.67·25-s − 6.31·26-s + 2.63·27-s + 3.06·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.26·3-s + 0.5·4-s − 1.23·5-s + 0.894·6-s + 1.16·7-s − 0.353·8-s + 0.599·9-s + 0.876·10-s − 1.28·11-s − 0.632·12-s + 1.75·13-s − 0.820·14-s + 1.56·15-s + 0.250·16-s − 1.37·17-s − 0.423·18-s + 0.699·19-s − 0.619·20-s − 1.46·21-s + 0.911·22-s − 0.659·23-s + 0.447·24-s + 0.535·25-s − 1.23·26-s + 0.506·27-s + 0.580·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4458850818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4458850818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.19T + 3T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 - 6.31T + 13T^{2} \) |
| 17 | \( 1 + 5.67T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + 5.56T + 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 - 0.630T + 37T^{2} \) |
| 41 | \( 1 - 8.42T + 41T^{2} \) |
| 43 | \( 1 + 9.98T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 - 6.71T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 + 5.77T + 67T^{2} \) |
| 71 | \( 1 - 4.19T + 71T^{2} \) |
| 73 | \( 1 + 6.91T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 4.21T + 83T^{2} \) |
| 89 | \( 1 + 5.49T + 89T^{2} \) |
| 97 | \( 1 - 5.15T + 97T^{2} \) |
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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
−7.76063485808317262214317907743, −7.37023484301631567606230173342, −6.45798152805911170256141484755, −5.75988755289669476202875509193, −5.16014299343251137950353956473, −4.35497496324985880635619152990, −3.68458308702951984361469766928, −2.47768929744941206509136960234, −1.38491617979123400416430242832, −0.42266555391287047240355416201,
0.42266555391287047240355416201, 1.38491617979123400416430242832, 2.47768929744941206509136960234, 3.68458308702951984361469766928, 4.35497496324985880635619152990, 5.16014299343251137950353956473, 5.75988755289669476202875509193, 6.45798152805911170256141484755, 7.37023484301631567606230173342, 7.76063485808317262214317907743
