| L(s) = 1 | − 2.38·2-s − 0.731·3-s + 3.69·4-s + 1.80·5-s + 1.74·6-s − 2.93·7-s − 4.04·8-s − 2.46·9-s − 4.29·10-s + 0.347·11-s − 2.70·12-s − 1.80·13-s + 7.00·14-s − 1.31·15-s + 2.26·16-s + 17-s + 5.88·18-s − 1.44·19-s + 6.65·20-s + 2.14·21-s − 0.830·22-s − 1.89·23-s + 2.95·24-s − 1.75·25-s + 4.30·26-s + 3.99·27-s − 10.8·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.422·3-s + 1.84·4-s + 0.805·5-s + 0.712·6-s − 1.10·7-s − 1.42·8-s − 0.821·9-s − 1.35·10-s + 0.104·11-s − 0.779·12-s − 0.500·13-s + 1.87·14-s − 0.340·15-s + 0.565·16-s + 0.242·17-s + 1.38·18-s − 0.332·19-s + 1.48·20-s + 0.468·21-s − 0.177·22-s − 0.395·23-s + 0.603·24-s − 0.351·25-s + 0.844·26-s + 0.769·27-s − 2.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 353 | \( 1 - T \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 3 | \( 1 + 0.731T + 3T^{2} \) |
| 5 | \( 1 - 1.80T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 - 0.347T + 11T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 19 | \( 1 + 1.44T + 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 5.61T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 - 4.45T + 53T^{2} \) |
| 59 | \( 1 - 9.13T + 59T^{2} \) |
| 61 | \( 1 + 4.99T + 61T^{2} \) |
| 67 | \( 1 - 3.64T + 67T^{2} \) |
| 71 | \( 1 + 1.83T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 8.20T + 79T^{2} \) |
| 83 | \( 1 + 1.21T + 83T^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.84347719029223870447963310418, −7.13362148587809791834854642526, −6.28394145015540029624821636370, −6.06612507790815876704601153327, −5.15143203331663564880906128029, −3.86148639145333451350145064519, −2.69464100090657212708861825721, −2.22212223933363656901538650866, −0.944353145030381868729647524157, 0,
0.944353145030381868729647524157, 2.22212223933363656901538650866, 2.69464100090657212708861825721, 3.86148639145333451350145064519, 5.15143203331663564880906128029, 6.06612507790815876704601153327, 6.28394145015540029624821636370, 7.13362148587809791834854642526, 7.84347719029223870447963310418
