L(s) = 1 | + (0.281 − 0.959i)7-s + (0.755 + 0.654i)11-s + (−0.959 + 0.281i)13-s + (0.989 − 0.142i)17-s + (−0.989 − 0.142i)19-s + (−0.989 + 0.142i)29-s + (0.841 − 0.540i)31-s + (0.415 + 0.909i)37-s + (0.415 − 0.909i)41-s + (−0.841 − 0.540i)43-s + i·47-s + (−0.841 − 0.540i)49-s + (−0.959 − 0.281i)53-s + (−0.281 − 0.959i)59-s + (−0.540 − 0.841i)61-s + ⋯ |
L(s) = 1 | + (0.281 − 0.959i)7-s + (0.755 + 0.654i)11-s + (−0.959 + 0.281i)13-s + (0.989 − 0.142i)17-s + (−0.989 − 0.142i)19-s + (−0.989 + 0.142i)29-s + (0.841 − 0.540i)31-s + (0.415 + 0.909i)37-s + (0.415 − 0.909i)41-s + (−0.841 − 0.540i)43-s + i·47-s + (−0.841 − 0.540i)49-s + (−0.959 − 0.281i)53-s + (−0.281 − 0.959i)59-s + (−0.540 − 0.841i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8115014221 - 0.9998456408i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8115014221 - 0.9998456408i\) |
\(L(1)\) |
\(\approx\) |
\(1.006604461 - 0.1709434785i\) |
\(L(1)\) |
\(\approx\) |
\(1.006604461 - 0.1709434785i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + (0.281 - 0.959i)T \) |
| 11 | \( 1 + (0.755 + 0.654i)T \) |
| 13 | \( 1 + (-0.959 + 0.281i)T \) |
| 17 | \( 1 + (0.989 - 0.142i)T \) |
| 19 | \( 1 + (-0.989 - 0.142i)T \) |
| 29 | \( 1 + (-0.989 + 0.142i)T \) |
| 31 | \( 1 + (0.841 - 0.540i)T \) |
| 37 | \( 1 + (0.415 + 0.909i)T \) |
| 41 | \( 1 + (0.415 - 0.909i)T \) |
| 43 | \( 1 + (-0.841 - 0.540i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.959 - 0.281i)T \) |
| 59 | \( 1 + (-0.281 - 0.959i)T \) |
| 61 | \( 1 + (-0.540 - 0.841i)T \) |
| 67 | \( 1 + (0.654 + 0.755i)T \) |
| 71 | \( 1 + (-0.654 - 0.755i)T \) |
| 73 | \( 1 + (0.989 + 0.142i)T \) |
| 79 | \( 1 + (0.959 - 0.281i)T \) |
| 83 | \( 1 + (-0.415 - 0.909i)T \) |
| 89 | \( 1 + (-0.841 - 0.540i)T \) |
| 97 | \( 1 + (-0.909 - 0.415i)T \) |
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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
−18.15755928798811153302745084084, −17.28642014419126864617947058438, −16.77486529750280113136630111433, −16.20664880995027510526178694609, −15.14453130221422308387152598717, −14.87446521808544734555551121592, −14.257264601858304410148356034751, −13.44338864478861448942433682347, −12.47747858001043415978879978315, −12.25074685322353042545748004448, −11.41556945088539446640997443727, −10.84061378882690076949478579751, −9.8614410573484892745120583268, −9.40446986992291253904252042918, −8.53276049518418217185515486187, −8.0839805434540451592016131528, −7.249683587888975225782326566935, −6.326323468601715398246006664789, −5.80289663648237022861501190267, −5.08841303054721832861253298823, −4.30122594930928562846182192588, −3.41008797375656827232031897173, −2.68350735419115597718535572364, −1.90505761631685925328151411566, −1.013913567269440932552774828476,
0.350696897325340516260067063593, 1.44557160034685231945018454838, 2.06751367923929282567811174803, 3.117264483646116017862502653597, 3.9483533250553331623834214040, 4.55026270193241165623195790984, 5.15199979661789054191214372127, 6.26400195033422743296833389715, 6.81548169366337321554010318310, 7.5744202310612899477689852659, 8.015673637303708795653487020314, 9.06890980156392930241671819774, 9.74121444594193688156471325478, 10.19548655004582460432491367980, 11.057707416888610748287751004614, 11.68032544757168664661689065514, 12.40783885415789691453897128189, 12.96579423703008129411920535651, 13.87894367452976927189411921171, 14.40755318711292372816336485618, 14.89101007968863655693772057604, 15.641035805180768341292018246169, 16.66075466634092816335713947101, 17.14279339596663841331490662743, 17.30010428175723049189729045481