L(s) = 1 | + (0.845 − 0.534i)2-s + (0.428 − 0.903i)4-s + (0.987 + 0.160i)5-s + (−0.120 − 0.992i)8-s + (0.919 − 0.391i)10-s + (0.845 + 0.534i)11-s + (−0.632 − 0.774i)16-s + (0.799 − 0.600i)17-s + (0.5 − 0.866i)19-s + (0.568 − 0.822i)20-s + 22-s + (0.5 − 0.866i)23-s + (0.948 + 0.316i)25-s + (−0.885 − 0.464i)29-s + (0.200 − 0.979i)31-s + (−0.948 − 0.316i)32-s + ⋯ |
L(s) = 1 | + (0.845 − 0.534i)2-s + (0.428 − 0.903i)4-s + (0.987 + 0.160i)5-s + (−0.120 − 0.992i)8-s + (0.919 − 0.391i)10-s + (0.845 + 0.534i)11-s + (−0.632 − 0.774i)16-s + (0.799 − 0.600i)17-s + (0.5 − 0.866i)19-s + (0.568 − 0.822i)20-s + 22-s + (0.5 − 0.866i)23-s + (0.948 + 0.316i)25-s + (−0.885 − 0.464i)29-s + (0.200 − 0.979i)31-s + (−0.948 − 0.316i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0183 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0183 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.898748305 - 2.846144340i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.898748305 - 2.846144340i\) |
\(L(1)\) |
\(\approx\) |
\(1.976642969 - 0.9541147737i\) |
\(L(1)\) |
\(\approx\) |
\(1.976642969 - 0.9541147737i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.845 - 0.534i)T \) |
| 5 | \( 1 + (0.987 + 0.160i)T \) |
| 11 | \( 1 + (0.845 + 0.534i)T \) |
| 17 | \( 1 + (0.799 - 0.600i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.885 - 0.464i)T \) |
| 31 | \( 1 + (0.200 - 0.979i)T \) |
| 37 | \( 1 + (0.948 - 0.316i)T \) |
| 41 | \( 1 + (-0.970 - 0.239i)T \) |
| 43 | \( 1 + (-0.748 + 0.663i)T \) |
| 47 | \( 1 + (0.428 + 0.903i)T \) |
| 53 | \( 1 + (-0.799 + 0.600i)T \) |
| 59 | \( 1 + (-0.632 + 0.774i)T \) |
| 61 | \( 1 + (-0.799 - 0.600i)T \) |
| 67 | \( 1 + (-0.996 - 0.0804i)T \) |
| 71 | \( 1 + (0.970 + 0.239i)T \) |
| 73 | \( 1 + (0.845 + 0.534i)T \) |
| 79 | \( 1 + (0.428 + 0.903i)T \) |
| 83 | \( 1 + (-0.970 + 0.239i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.354 - 0.935i)T \) |
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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
−18.754386077069697276917524802693, −18.08295638589828974363586820559, −17.17261692507212196126022042974, −16.79080121621636388034060484505, −16.32029413206708469753880101198, −15.24676559055149966557557893039, −14.673647229970312066545869318448, −14.00500163596609666366062037428, −13.56210476129657065839897636076, −12.77825463702626555287271506280, −12.12997632422419995268267456816, −11.44386831151279168751475871682, −10.55925562154003056188895281908, −9.73182085042511739164031270555, −8.94557192655435186080969774089, −8.25721453932173313010874718585, −7.390189787817211895541070993438, −6.58875082985475968322857902889, −5.94156210047493772925184982171, −5.404865064949051736133061434864, −4.67480788069297674827542386240, −3.43572405500250043627755242052, −3.30534934924998542774743624934, −1.88997095471600292396860991068, −1.34402211700076922439633082132,
0.87776853526099349692524460078, 1.64245525453453695332991583500, 2.538667495546377263767906240735, 3.08659186812690104634830078506, 4.156130327872209386735935093613, 4.83671144826422453479043392204, 5.56925664376357774619739544794, 6.31087406679350328094614537165, 6.89175637917752842453360180976, 7.739981298930138936993597576858, 9.21035641037773559539472808973, 9.46210550871152081996238170392, 10.169647212308162665451364415810, 11.05335672391863782875972491321, 11.58120738684509484380529049794, 12.44342041593075824088327238675, 13.00136694382888413660818328972, 13.789420172550225708493276143111, 14.21535612869087743340708620795, 14.959704738714494505682502776273, 15.48183958988847055519835472298, 16.69764013762621059781435778439, 16.9670207361541344595773526768, 18.110882673798124650861373361084, 18.537572044805757843630106892882