L(s) = 1 | + (−0.924 − 0.380i)2-s + (0.743 − 0.669i)3-s + (0.710 + 0.703i)4-s + (−0.941 + 0.336i)6-s + (−0.389 − 0.921i)8-s + (0.104 − 0.994i)9-s + (0.998 + 0.0475i)12-s + (0.931 − 0.362i)13-s + (0.00951 + 0.999i)16-s + (0.244 + 0.969i)17-s + (−0.475 + 0.879i)18-s + (−0.0665 − 0.997i)19-s + (−0.814 − 0.580i)23-s + (−0.905 − 0.424i)24-s + (−0.999 − 0.0190i)26-s + (−0.587 − 0.809i)27-s + ⋯ |
L(s) = 1 | + (−0.924 − 0.380i)2-s + (0.743 − 0.669i)3-s + (0.710 + 0.703i)4-s + (−0.941 + 0.336i)6-s + (−0.389 − 0.921i)8-s + (0.104 − 0.994i)9-s + (0.998 + 0.0475i)12-s + (0.931 − 0.362i)13-s + (0.00951 + 0.999i)16-s + (0.244 + 0.969i)17-s + (−0.475 + 0.879i)18-s + (−0.0665 − 0.997i)19-s + (−0.814 − 0.580i)23-s + (−0.905 − 0.424i)24-s + (−0.999 − 0.0190i)26-s + (−0.587 − 0.809i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5209976625 - 1.295109688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5209976625 - 1.295109688i\) |
\(L(1)\) |
\(\approx\) |
\(0.8156962791 - 0.4697931902i\) |
\(L(1)\) |
\(\approx\) |
\(0.8156962791 - 0.4697931902i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.924 - 0.380i)T \) |
| 3 | \( 1 + (0.743 - 0.669i)T \) |
| 13 | \( 1 + (0.931 - 0.362i)T \) |
| 17 | \( 1 + (0.244 + 0.969i)T \) |
| 19 | \( 1 + (-0.0665 - 0.997i)T \) |
| 23 | \( 1 + (-0.814 - 0.580i)T \) |
| 29 | \( 1 + (-0.985 + 0.170i)T \) |
| 31 | \( 1 + (0.548 - 0.836i)T \) |
| 37 | \( 1 + (0.986 + 0.161i)T \) |
| 41 | \( 1 + (0.564 - 0.825i)T \) |
| 43 | \( 1 + (0.755 - 0.654i)T \) |
| 47 | \( 1 + (0.475 + 0.879i)T \) |
| 53 | \( 1 + (-0.999 - 0.00951i)T \) |
| 59 | \( 1 + (-0.997 + 0.0760i)T \) |
| 61 | \( 1 + (0.991 + 0.132i)T \) |
| 67 | \( 1 + (0.690 + 0.723i)T \) |
| 71 | \( 1 + (-0.466 - 0.884i)T \) |
| 73 | \( 1 + (0.976 - 0.217i)T \) |
| 79 | \( 1 + (-0.683 - 0.730i)T \) |
| 83 | \( 1 + (-0.980 + 0.198i)T \) |
| 89 | \( 1 + (0.786 - 0.618i)T \) |
| 97 | \( 1 + (-0.996 + 0.0855i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.61894357635861115287541739420, −18.105572204363894626709651153176, −17.162926774780237520589000094731, −16.40631576096484035891699454055, −16.025575184944867144229822507231, −15.48453473341053574717574174345, −14.589384139822152585407959976556, −14.15504676133268323573448846309, −13.49993064718859571314203562180, −12.428229869636717507894458934262, −11.39700593782765896401486233085, −11.02916536764913252849888475697, −10.07449852597944408332772398479, −9.60380527740118687887672239370, −9.05963966816766455984319202774, −8.16307345808062825693483045291, −7.85096900615135570946327189057, −6.969901209619566725820270665793, −6.03504642067723520699844407520, −5.4242132434552878514697227486, −4.43037606073117525332003940435, −3.609711772620074373229380437555, −2.77274051685342911681953345442, −1.91898987571737348921502143694, −1.12305883455728591427525220864,
0.50542170134163638012023702474, 1.33072115629467209840147443748, 2.15561223660720757181024949663, 2.78513299073519263408096994419, 3.67173202638000384678478609826, 4.247049502386859430389609359093, 5.907342078941804685258965406707, 6.29703330072438636671694651873, 7.25476226872888254093405919087, 7.856598351183300169632382435628, 8.41471162825945715029142122285, 9.07250154406737200548904796532, 9.66725042297041607104392156233, 10.5850133684250990840932525918, 11.150893170988618857597702756740, 11.96657892754480120748863415343, 12.75683213235869099617126961000, 13.10064496953420163667028844484, 13.9673101952317469086002844275, 14.81702973967589373562853951352, 15.51587150666177617807458290187, 16.05022469210670404250825049933, 17.11975813830894288470533710224, 17.51717463056016332991841469913, 18.30970036580107742388023846366