| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 11-s + 12-s − 13-s + 14-s + 16-s + 17-s + 18-s − 19-s + 21-s + 22-s − 23-s + 24-s − 26-s + 27-s + 28-s − 29-s − 31-s + 32-s + 33-s + 34-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 11-s + 12-s − 13-s + 14-s + 16-s + 17-s + 18-s − 19-s + 21-s + 22-s − 23-s + 24-s − 26-s + 27-s + 28-s − 29-s − 31-s + 32-s + 33-s + 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1345 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1345 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.120139643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.120139643\) |
| \(L(1)\) |
\(\approx\) |
\(3.003096456\) |
| \(L(1)\) |
\(\approx\) |
\(3.003096456\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 269 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 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
−20.82256637490439832211821028743, −20.38970093165888788972002739810, −19.50947023253359241684850723535, −19.03619450967851023891886737488, −17.826109022012622398383801837173, −16.904288660024443223951322860934, −16.21046347815937142514573763870, −15.01751340387213018189870514037, −14.66258370519443072054939616788, −14.26942131400414536680862204992, −13.37796274669708448454602577219, −12.43526014475053676071380269950, −11.93500437534040964355846842802, −10.930163120600492000761472541588, −10.061403611905611755792043931980, −9.15228697485704658491113515418, −8.080049620648697579815595238929, −7.53566753328879750814684379744, −6.68755175245097844772012771294, −5.59677473705351193930531788521, −4.65054906710023790694652322047, −3.9841799870421539420267301412, −3.17721956860378683471293504168, −1.99636948524670912706735356162, −1.60225696729338828828318851130,
1.60225696729338828828318851130, 1.99636948524670912706735356162, 3.17721956860378683471293504168, 3.9841799870421539420267301412, 4.65054906710023790694652322047, 5.59677473705351193930531788521, 6.68755175245097844772012771294, 7.53566753328879750814684379744, 8.080049620648697579815595238929, 9.15228697485704658491113515418, 10.061403611905611755792043931980, 10.930163120600492000761472541588, 11.93500437534040964355846842802, 12.43526014475053676071380269950, 13.37796274669708448454602577219, 14.26942131400414536680862204992, 14.66258370519443072054939616788, 15.01751340387213018189870514037, 16.21046347815937142514573763870, 16.904288660024443223951322860934, 17.826109022012622398383801837173, 19.03619450967851023891886737488, 19.50947023253359241684850723535, 20.38970093165888788972002739810, 20.82256637490439832211821028743
