| L(s) = 1 | − 2.14·2-s + 0.825·3-s + 2.59·4-s − 1.77·6-s + 2.42·7-s − 1.27·8-s − 2.31·9-s − 11-s + 2.14·12-s + 2.72·13-s − 5.19·14-s − 2.45·16-s + 17-s + 4.96·18-s − 3.54·19-s + 1.99·21-s + 2.14·22-s − 3.87·23-s − 1.05·24-s − 5.83·26-s − 4.39·27-s + 6.28·28-s − 0.0884·29-s + 0.556·31-s + 7.81·32-s − 0.825·33-s − 2.14·34-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 0.476·3-s + 1.29·4-s − 0.722·6-s + 0.915·7-s − 0.451·8-s − 0.772·9-s − 0.301·11-s + 0.618·12-s + 0.754·13-s − 1.38·14-s − 0.613·16-s + 0.242·17-s + 1.17·18-s − 0.812·19-s + 0.436·21-s + 0.457·22-s − 0.807·23-s − 0.215·24-s − 1.14·26-s − 0.845·27-s + 1.18·28-s − 0.0164·29-s + 0.0998·31-s + 1.38·32-s − 0.143·33-s − 0.367·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9672042447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9672042447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 3 | \( 1 - 0.825T + 3T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 13 | \( 1 - 2.72T + 13T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 + 0.0884T + 29T^{2} \) |
| 31 | \( 1 - 0.556T + 31T^{2} \) |
| 37 | \( 1 + 4.66T + 37T^{2} \) |
| 41 | \( 1 - 6.78T + 41T^{2} \) |
| 43 | \( 1 + 5.00T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 8.06T + 59T^{2} \) |
| 61 | \( 1 - 4.01T + 61T^{2} \) |
| 67 | \( 1 + 6.36T + 67T^{2} \) |
| 71 | \( 1 - 6.57T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 3.06T + 79T^{2} \) |
| 83 | \( 1 - 1.67T + 83T^{2} \) |
| 89 | \( 1 - 4.84T + 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
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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
−8.409780239909581896267549686675, −7.906666192528478367659703403262, −7.27848052915149240449030410198, −6.29925493648441861243516992039, −5.56403046074972738691721817147, −4.53542614904040177021173373210, −3.60091381629317638322168150043, −2.42396856058554641061738461066, −1.82941391073245174933699878797, −0.66862565924532468246240724115,
0.66862565924532468246240724115, 1.82941391073245174933699878797, 2.42396856058554641061738461066, 3.60091381629317638322168150043, 4.53542614904040177021173373210, 5.56403046074972738691721817147, 6.29925493648441861243516992039, 7.27848052915149240449030410198, 7.906666192528478367659703403262, 8.409780239909581896267549686675
