| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 17-s + 18-s − 20-s + 21-s + 22-s − 23-s + 24-s + 25-s − 26-s + 27-s + 28-s − 29-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 17-s + 18-s − 20-s + 21-s + 22-s − 23-s + 24-s + 25-s − 26-s + 27-s + 28-s − 29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(7.213609892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.213609892\) |
| \(L(1)\) |
\(\approx\) |
\(2.969999608\) |
| \(L(1)\) |
\(\approx\) |
\(2.969999608\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| 53 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.3809600261643226576404393234, −20.65154364704046416340154140236, −20.01731187106024450542901004502, −19.39236541447543458880568691224, −18.71874482222805838435677294310, −17.3355001775096218957706108914, −16.502754866877476188306248613818, −15.52220235786127297011234573777, −14.98636290411520308933372961431, −14.221685625778833584468104061775, −13.97486402430997061670300263004, −12.43491654969928414927929762750, −12.20052910147104792081140175549, −11.28770899323217625266602752018, −10.32848153379036246102758319686, −9.24675713623227745945491250192, −8.08596003349658017728818474882, −7.63013747176207567339015280377, −6.89863023115022546465676298367, −5.562830906135186847937045833159, −4.44772252504951939539336535757, −4.04260042206395130069883425231, −3.09315670436350488543574131204, −2.09461473063772594602013561200, −1.110091433171948066531185273272,
1.110091433171948066531185273272, 2.09461473063772594602013561200, 3.09315670436350488543574131204, 4.04260042206395130069883425231, 4.44772252504951939539336535757, 5.562830906135186847937045833159, 6.89863023115022546465676298367, 7.63013747176207567339015280377, 8.08596003349658017728818474882, 9.24675713623227745945491250192, 10.32848153379036246102758319686, 11.28770899323217625266602752018, 12.20052910147104792081140175549, 12.43491654969928414927929762750, 13.97486402430997061670300263004, 14.221685625778833584468104061775, 14.98636290411520308933372961431, 15.52220235786127297011234573777, 16.502754866877476188306248613818, 17.3355001775096218957706108914, 18.71874482222805838435677294310, 19.39236541447543458880568691224, 20.01731187106024450542901004502, 20.65154364704046416340154140236, 21.3809600261643226576404393234
