L(s) = 1 | + 2.30·2-s + 3.30·4-s + 5-s + 7-s + 3.00·8-s + 2.30·10-s + 3·11-s + 1.30·13-s + 2.30·14-s + 0.302·16-s − 2.30·17-s − 3.30·19-s + 3.30·20-s + 6.90·22-s + 0.697·23-s + 25-s + 3·26-s + 3.30·28-s + 5.30·29-s − 2.39·31-s − 5.30·32-s − 5.30·34-s + 35-s + 3.60·37-s − 7.60·38-s + 3.00·40-s − 6.90·41-s + ⋯ |
L(s) = 1 | + 1.62·2-s + 1.65·4-s + 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.728·10-s + 0.904·11-s + 0.361·13-s + 0.615·14-s + 0.0756·16-s − 0.558·17-s − 0.757·19-s + 0.738·20-s + 1.47·22-s + 0.145·23-s + 0.200·25-s + 0.588·26-s + 0.624·28-s + 0.984·29-s − 0.430·31-s − 0.937·32-s − 0.909·34-s + 0.169·35-s + 0.592·37-s − 1.23·38-s + 0.474·40-s − 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.399362437\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.399362437\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 + 3.30T + 19T^{2} \) |
| 23 | \( 1 - 0.697T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 + 6.90T + 41T^{2} \) |
| 43 | \( 1 + 4.21T + 43T^{2} \) |
| 47 | \( 1 - 7.60T + 47T^{2} \) |
| 53 | \( 1 + 5.51T + 53T^{2} \) |
| 59 | \( 1 - 6.21T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 2.09T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 9.51T + 97T^{2} \) |
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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
−10.32311307127426569087854726205, −9.168744348213449786245213795433, −8.379419098498571764255089098175, −7.00208684727925169722721474597, −6.41516095465803900924521091338, −5.61599385168131930874559514651, −4.64584271661518399921731908072, −3.97193850209979659721996294033, −2.83922855426164383936617844541, −1.69463971121385263586712241401,
1.69463971121385263586712241401, 2.83922855426164383936617844541, 3.97193850209979659721996294033, 4.64584271661518399921731908072, 5.61599385168131930874559514651, 6.41516095465803900924521091338, 7.00208684727925169722721474597, 8.379419098498571764255089098175, 9.168744348213449786245213795433, 10.32311307127426569087854726205