| L(s) = 1 | + 1.13·2-s − 10.3·3-s − 6.71·4-s − 20.0·5-s − 11.7·6-s − 16.6·8-s + 80.1·9-s − 22.7·10-s + 41.5·11-s + 69.4·12-s − 31.5·13-s + 207.·15-s + 34.7·16-s − 27.7·17-s + 90.8·18-s + 16.4·19-s + 134.·20-s + 47.1·22-s + 34.4·23-s + 172.·24-s + 275.·25-s − 35.7·26-s − 549.·27-s − 108.·29-s + 235.·30-s + 154.·31-s + 172.·32-s + ⋯ |
| L(s) = 1 | + 0.400·2-s − 1.99·3-s − 0.839·4-s − 1.79·5-s − 0.798·6-s − 0.737·8-s + 2.96·9-s − 0.718·10-s + 1.14·11-s + 1.67·12-s − 0.672·13-s + 3.56·15-s + 0.543·16-s − 0.396·17-s + 1.18·18-s + 0.198·19-s + 1.50·20-s + 0.457·22-s + 0.311·23-s + 1.46·24-s + 2.20·25-s − 0.269·26-s − 3.91·27-s − 0.693·29-s + 1.43·30-s + 0.895·31-s + 0.955·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1997977868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1997977868\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - 41T \) |
| good | 2 | \( 1 - 1.13T + 8T^{2} \) |
| 3 | \( 1 + 10.3T + 27T^{2} \) |
| 5 | \( 1 + 20.0T + 125T^{2} \) |
| 11 | \( 1 - 41.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 34.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 416.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 314.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 15.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 113.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 144.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 405.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 531.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 204.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 603.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 191.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 194.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 541.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.817168479409004257400250148225, −7.80077374179093971453029830621, −6.98638787964855909223878996547, −6.45993199814610760741552331759, −5.32317627679499461501403085174, −4.76972929589749052633600265679, −4.13119523770195837219998031597, −3.52317994561836823074766469880, −1.23599573912225812979519144918, −0.25280420569361265221922536640,
0.25280420569361265221922536640, 1.23599573912225812979519144918, 3.52317994561836823074766469880, 4.13119523770195837219998031597, 4.76972929589749052633600265679, 5.32317627679499461501403085174, 6.45993199814610760741552331759, 6.98638787964855909223878996547, 7.80077374179093971453029830621, 8.817168479409004257400250148225
