| L(s) = 1 | − 2.73·2-s − 3-s + 5.49·4-s − 3.81·5-s + 2.73·6-s − 1.31·7-s − 9.55·8-s + 9-s + 10.4·10-s − 0.116·11-s − 5.49·12-s + 4.88·13-s + 3.59·14-s + 3.81·15-s + 15.1·16-s + 17-s − 2.73·18-s − 0.495·19-s − 20.9·20-s + 1.31·21-s + 0.318·22-s + 2.52·23-s + 9.55·24-s + 9.56·25-s − 13.3·26-s − 27-s − 7.21·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 0.577·3-s + 2.74·4-s − 1.70·5-s + 1.11·6-s − 0.496·7-s − 3.37·8-s + 0.333·9-s + 3.30·10-s − 0.0350·11-s − 1.58·12-s + 1.35·13-s + 0.960·14-s + 0.985·15-s + 3.79·16-s + 0.242·17-s − 0.645·18-s − 0.113·19-s − 4.68·20-s + 0.286·21-s + 0.0677·22-s + 0.526·23-s + 1.95·24-s + 1.91·25-s − 2.62·26-s − 0.192·27-s − 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3671952143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3671952143\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 + 3.81T + 5T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 + 0.116T + 11T^{2} \) |
| 13 | \( 1 - 4.88T + 13T^{2} \) |
| 19 | \( 1 + 0.495T + 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 - 8.50T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 4.65T + 37T^{2} \) |
| 41 | \( 1 + 5.83T + 41T^{2} \) |
| 43 | \( 1 + 7.57T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 + 8.29T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 7.42T + 67T^{2} \) |
| 71 | \( 1 + 1.93T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 83 | \( 1 + 6.08T + 83T^{2} \) |
| 89 | \( 1 + 2.64T + 89T^{2} \) |
| 97 | \( 1 - 4.33T + 97T^{2} \) |
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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
−8.557683169802929387180912488410, −7.930136404806652457250198911571, −7.09386560892295468335093399181, −6.66223514999318078277584989903, −5.94968704223758402861016582977, −4.61707797786979028482383085327, −3.51504057823178788600330084724, −2.87157309142734664477894151130, −1.28166580761734409663625852502, −0.54420626546707296392817011453,
0.54420626546707296392817011453, 1.28166580761734409663625852502, 2.87157309142734664477894151130, 3.51504057823178788600330084724, 4.61707797786979028482383085327, 5.94968704223758402861016582977, 6.66223514999318078277584989903, 7.09386560892295468335093399181, 7.930136404806652457250198911571, 8.557683169802929387180912488410
