| L(s) = 1 | + (0.669 − 0.743i)3-s + (−0.994 − 0.104i)5-s + (−0.104 − 0.994i)9-s + (−0.743 + 0.669i)15-s + (−0.809 − 0.587i)17-s + (−0.207 + 0.978i)19-s + 23-s + (0.978 + 0.207i)25-s + (−0.809 − 0.587i)27-s + (0.669 + 0.743i)29-s + (−0.994 + 0.104i)31-s + (0.951 + 0.309i)37-s + (0.207 − 0.978i)41-s + (−0.5 − 0.866i)43-s + i·45-s + ⋯ |
| L(s) = 1 | + (0.669 − 0.743i)3-s + (−0.994 − 0.104i)5-s + (−0.104 − 0.994i)9-s + (−0.743 + 0.669i)15-s + (−0.809 − 0.587i)17-s + (−0.207 + 0.978i)19-s + 23-s + (0.978 + 0.207i)25-s + (−0.809 − 0.587i)27-s + (0.669 + 0.743i)29-s + (−0.994 + 0.104i)31-s + (0.951 + 0.309i)37-s + (0.207 − 0.978i)41-s + (−0.5 − 0.866i)43-s + i·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1302432103 - 0.8530470264i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.1302432103 - 0.8530470264i\) |
| \(L(1)\) |
\(\approx\) |
\(0.9420173051 - 0.3529057216i\) |
| \(L(1)\) |
\(\approx\) |
\(0.9420173051 - 0.3529057216i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (-0.994 - 0.104i)T \) |
| 17 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.207 + 0.978i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.669 + 0.743i)T \) |
| 31 | \( 1 + (-0.994 + 0.104i)T \) |
| 37 | \( 1 + (0.951 + 0.309i)T \) |
| 41 | \( 1 + (0.207 - 0.978i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.743 + 0.669i)T \) |
| 53 | \( 1 + (-0.104 - 0.994i)T \) |
| 59 | \( 1 + (0.951 + 0.309i)T \) |
| 61 | \( 1 + (-0.913 + 0.406i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.994 - 0.104i)T \) |
| 73 | \( 1 + (0.743 - 0.669i)T \) |
| 79 | \( 1 + (-0.913 - 0.406i)T \) |
| 83 | \( 1 + (-0.587 + 0.809i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (-0.406 + 0.913i)T \) |
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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
−18.88651724342456973781418789808, −18.09484476847821302290726077913, −17.13245969819406252832679005095, −16.55148067986896491339281780748, −15.74726642507706028068187836198, −15.29677910818423320831576488690, −14.82259439928715572876743378782, −14.078911589737083034906931143308, −13.12059641325019539917371567665, −12.76390491411611831586352511653, −11.45416802055974485853860695581, −11.20689818508407359892737407921, −10.46190351295373742435402192706, −9.60414658984861174411571520636, −8.84673644750853857373674338580, −8.3995620378418370596391924537, −7.58017415501677977867274063850, −6.949751906076959469587472573337, −5.98212351472934476560720698578, −4.83246385391585493147357624988, −4.437030354255480242362614027295, −3.695282536052584356448845863212, −2.89803801651844633201048480061, −2.26428494464452931879259153203, −0.96096773041096410501222505545,
0.14158053880734206251981455057, 0.952750815307441581161122765294, 1.87921839745230197494433851223, 2.79772736346616661580512805387, 3.48409732097260564763383603052, 4.203505484039170509408174862759, 5.07561380572167948181842855106, 6.0862090396830508486817616362, 7.03231829465678783731692154928, 7.30405989914476034282809437258, 8.20525654741547999460979290633, 8.7526928445277609833818403181, 9.32672439978243780654138078729, 10.41783565301687176549915818072, 11.19535769834611768649921196630, 11.8657859123167243694315305177, 12.549377483375823126749896069363, 13.02469395216602427348031894031, 13.8852373198397470184828995165, 14.55793298419017009047896053088, 15.126902128598045650698764029616, 15.81321155490516110090468865145, 16.52661710791728782950371392100, 17.3163096214984523030958736521, 18.22359857805782309166958416424
