| L(s) = 1 | − 0.256·2-s + 0.196·3-s − 1.93·4-s + 5-s − 0.0502·6-s + 2.99·7-s + 1.00·8-s − 2.96·9-s − 0.256·10-s − 4.51·11-s − 0.380·12-s + 1.63·13-s − 0.766·14-s + 0.196·15-s + 3.61·16-s + 17-s + 0.758·18-s − 3.34·19-s − 1.93·20-s + 0.588·21-s + 1.15·22-s + 4.80·23-s + 0.197·24-s + 25-s − 0.418·26-s − 1.17·27-s − 5.79·28-s + ⋯ |
| L(s) = 1 | − 0.181·2-s + 0.113·3-s − 0.967·4-s + 0.447·5-s − 0.0205·6-s + 1.13·7-s + 0.356·8-s − 0.987·9-s − 0.0809·10-s − 1.36·11-s − 0.109·12-s + 0.453·13-s − 0.204·14-s + 0.0507·15-s + 0.902·16-s + 0.242·17-s + 0.178·18-s − 0.767·19-s − 0.432·20-s + 0.128·21-s + 0.246·22-s + 1.00·23-s + 0.0403·24-s + 0.200·25-s − 0.0820·26-s − 0.225·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 + 0.256T + 2T^{2} \) |
| 3 | \( 1 - 0.196T + 3T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 + 4.51T + 11T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 19 | \( 1 + 3.34T + 19T^{2} \) |
| 23 | \( 1 - 4.80T + 23T^{2} \) |
| 29 | \( 1 + 5.40T + 29T^{2} \) |
| 31 | \( 1 - 0.0151T + 31T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 - 5.25T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 - 0.863T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 9.17T + 59T^{2} \) |
| 61 | \( 1 + 4.43T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 73 | \( 1 - 8.55T + 73T^{2} \) |
| 79 | \( 1 - 1.73T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2.45T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.030639643124293716696136752926, −7.27627508722263853586073088634, −6.01498271603267481907755169792, −5.48332379040242955544789221614, −4.93692683523541842506500940900, −4.20259676017485715956628886616, −3.14185679435922108753218889269, −2.31837882181507923329566356058, −1.24964638625248854470551247889, 0,
1.24964638625248854470551247889, 2.31837882181507923329566356058, 3.14185679435922108753218889269, 4.20259676017485715956628886616, 4.93692683523541842506500940900, 5.48332379040242955544789221614, 6.01498271603267481907755169792, 7.27627508722263853586073088634, 8.030639643124293716696136752926
