L(s) = 1 | + (0.959 + 0.281i)3-s + (−0.415 − 0.909i)5-s + (0.415 + 0.909i)7-s + (0.841 + 0.540i)9-s + (0.415 − 0.909i)11-s + (−0.959 − 0.281i)13-s + (−0.142 − 0.989i)15-s + (−0.142 + 0.989i)17-s + (−0.841 − 0.540i)19-s + (0.142 + 0.989i)21-s + (0.841 + 0.540i)23-s + (−0.654 + 0.755i)25-s + (0.654 + 0.755i)27-s + (0.415 + 0.909i)29-s + (0.841 − 0.540i)31-s + ⋯ |
L(s) = 1 | + (0.959 + 0.281i)3-s + (−0.415 − 0.909i)5-s + (0.415 + 0.909i)7-s + (0.841 + 0.540i)9-s + (0.415 − 0.909i)11-s + (−0.959 − 0.281i)13-s + (−0.142 − 0.989i)15-s + (−0.142 + 0.989i)17-s + (−0.841 − 0.540i)19-s + (0.142 + 0.989i)21-s + (0.841 + 0.540i)23-s + (−0.654 + 0.755i)25-s + (0.654 + 0.755i)27-s + (0.415 + 0.909i)29-s + (0.841 − 0.540i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.927675671 + 0.6550397055i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.927675671 + 0.6550397055i\) |
\(L(1)\) |
\(\approx\) |
\(1.509767635 + 0.1058094347i\) |
\(L(1)\) |
\(\approx\) |
\(1.509767635 + 0.1058094347i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (0.959 + 0.281i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 7 | \( 1 + (0.415 + 0.909i)T \) |
| 11 | \( 1 + (0.415 - 0.909i)T \) |
| 13 | \( 1 + (-0.959 - 0.281i)T \) |
| 17 | \( 1 + (-0.142 + 0.989i)T \) |
| 19 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (0.841 + 0.540i)T \) |
| 29 | \( 1 + (0.415 + 0.909i)T \) |
| 31 | \( 1 + (0.841 - 0.540i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.959 - 0.281i)T \) |
| 43 | \( 1 + (-0.415 + 0.909i)T \) |
| 47 | \( 1 + (0.959 - 0.281i)T \) |
| 53 | \( 1 + (0.959 + 0.281i)T \) |
| 59 | \( 1 + (0.959 - 0.281i)T \) |
| 61 | \( 1 + (-0.654 - 0.755i)T \) |
| 67 | \( 1 + (-0.959 - 0.281i)T \) |
| 71 | \( 1 + (-0.415 + 0.909i)T \) |
| 73 | \( 1 + (0.841 - 0.540i)T \) |
| 79 | \( 1 + (-0.841 + 0.540i)T \) |
| 83 | \( 1 + (0.142 - 0.989i)T \) |
| 97 | \( 1 + (0.415 + 0.909i)T \) |
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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
−22.53005816487207266082333745706, −21.312838979504278108971307481504, −20.6456159298198235200625111000, −19.7319737318140715383607992856, −19.32473901685543021577079312691, −18.36037516860011682415990686409, −17.594732631032271361493995192446, −16.68186419760467437827900381360, −15.434452849023043857834564127759, −14.75259622326248858170299734767, −14.27480642531285046469114833754, −13.45797470169801148285297804105, −12.35956300753480419178635357177, −11.59170630322932750813303795460, −10.396834924338951530772131999877, −9.850659935758132048923736240, −8.74251132012350363188567435942, −7.63277211587132987711095704150, −7.21207308298788989680019817372, −6.48455216377539844100636220856, −4.564461463420288343782760728042, −4.1141047571663211135854808919, −2.8353847071616599258713029410, −2.13120355219734356551826757279, −0.73841643809188202799503552899,
0.95615646520464397694070393519, 2.14563772368828694156338507363, 3.10511705889198619583596503275, 4.240529905031170667518503270888, 4.95793287689138624658335814004, 6.03003666568468803858963158654, 7.43097770635281710926226893794, 8.34156463426972031120120481381, 8.79173748254730213188928888721, 9.522079171353182865559963547515, 10.74557293532802189506355388905, 11.69549982860643269677030460160, 12.668132617023770613967253084270, 13.22917059553728183315868557872, 14.421644287329876829204891756814, 15.07540162664192985559838081840, 15.68368349407935461288607443366, 16.68830137407532864305001382514, 17.41069998115966732971908143879, 18.67086019852120021418150737816, 19.554684399851729585343545939290, 19.71802813167169526301768190458, 20.97752559341007137206194544536, 21.50571253507262483585621912414, 22.066127240019804248117791938117