L(s) = 1 | + (0.142 − 0.989i)7-s + (−0.909 + 0.415i)11-s + (0.989 − 0.142i)13-s + (−0.654 + 0.755i)17-s + (0.755 − 0.654i)19-s + (−0.755 − 0.654i)29-s + (0.959 − 0.281i)31-s + (0.540 − 0.841i)37-s + (0.841 − 0.540i)41-s + (0.281 − 0.959i)43-s − 47-s + (−0.959 − 0.281i)49-s + (−0.989 − 0.142i)53-s + (0.989 − 0.142i)59-s + (0.281 + 0.959i)61-s + ⋯ |
L(s) = 1 | + (0.142 − 0.989i)7-s + (−0.909 + 0.415i)11-s + (0.989 − 0.142i)13-s + (−0.654 + 0.755i)17-s + (0.755 − 0.654i)19-s + (−0.755 − 0.654i)29-s + (0.959 − 0.281i)31-s + (0.540 − 0.841i)37-s + (0.841 − 0.540i)41-s + (0.281 − 0.959i)43-s − 47-s + (−0.959 − 0.281i)49-s + (−0.989 − 0.142i)53-s + (0.989 − 0.142i)59-s + (0.281 + 0.959i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6823836885 - 1.067415319i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6823836885 - 1.067415319i\) |
\(L(1)\) |
\(\approx\) |
\(0.9808064863 - 0.2166244975i\) |
\(L(1)\) |
\(\approx\) |
\(0.9808064863 - 0.2166244975i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + (0.142 - 0.989i)T \) |
| 11 | \( 1 + (-0.909 + 0.415i)T \) |
| 13 | \( 1 + (0.989 - 0.142i)T \) |
| 17 | \( 1 + (-0.654 + 0.755i)T \) |
| 19 | \( 1 + (0.755 - 0.654i)T \) |
| 29 | \( 1 + (-0.755 - 0.654i)T \) |
| 31 | \( 1 + (0.959 - 0.281i)T \) |
| 37 | \( 1 + (0.540 - 0.841i)T \) |
| 41 | \( 1 + (0.841 - 0.540i)T \) |
| 43 | \( 1 + (0.281 - 0.959i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.989 - 0.142i)T \) |
| 59 | \( 1 + (0.989 - 0.142i)T \) |
| 61 | \( 1 + (0.281 + 0.959i)T \) |
| 67 | \( 1 + (-0.909 - 0.415i)T \) |
| 71 | \( 1 + (-0.415 + 0.909i)T \) |
| 73 | \( 1 + (-0.654 - 0.755i)T \) |
| 79 | \( 1 + (0.142 + 0.989i)T \) |
| 83 | \( 1 + (-0.540 + 0.841i)T \) |
| 89 | \( 1 + (-0.959 - 0.281i)T \) |
| 97 | \( 1 + (-0.841 + 0.540i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.069244922985142690845741989, −17.74104284664743728977195963177, −16.48749507692721920898754658645, −16.045208637233765996468686032280, −15.6296182810847818545466789741, −14.78314365655265744079298758976, −14.151233920936892893053012703539, −13.316695891815341262822311733586, −12.937828829011949274295147639314, −12.00910681950412358943754238275, −11.367245106485086062189248858882, −10.95751780209876112320226710210, −9.93692259138849665881647056733, −9.366664606642192305169193434585, −8.55464921712894949892193883991, −8.10906875635446056858442483731, −7.31655120451567296274765183546, −6.29712992777692797441591976071, −5.86423627367582439390001364618, −5.058796179153495516198056890718, −4.44990984840291157096024159027, −3.22199164990717204932061590233, −2.89530230405444157250066160323, −1.89272276461517825002106550834, −1.064348654266572383071474917977,
0.351900508371459159320424024772, 1.28162946835247070302708231408, 2.183718289688479197219834856197, 3.021833027039040850858063409696, 3.971446669240287497669791850016, 4.382819835101097478661036564996, 5.347601551247813682747085179430, 6.0244478179153480961244910198, 6.87938960279848444303982543422, 7.51656659024317619199190407312, 8.118581289766908607990177743, 8.85389472655229353006290767225, 9.75251332491018698380334890045, 10.316417409666661995546772274789, 11.0783155445262242897993579087, 11.37461699299622050871408668071, 12.53565662790987337133629738147, 13.17335560026716263249028405454, 13.5420790239467556533040901262, 14.27161323785506201221950774888, 15.15927936923799745531911601599, 15.67718463703817547249685180329, 16.28000530535164841380262882269, 17.06464183223233569665839543449, 17.858721966316784425308666124839