L(s) = 1 | + (0.984 + 0.173i)2-s + (−0.597 − 0.802i)3-s + (0.939 + 0.342i)4-s + (−0.727 − 0.686i)5-s + (−0.448 − 0.893i)6-s + (0.973 + 0.230i)7-s + (0.866 + 0.5i)8-s + (−0.286 + 0.957i)9-s + (−0.597 − 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.286 − 0.957i)12-s + (0.230 − 0.973i)13-s + (0.918 + 0.396i)14-s + (−0.116 + 0.993i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯ |
L(s) = 1 | + (0.984 + 0.173i)2-s + (−0.597 − 0.802i)3-s + (0.939 + 0.342i)4-s + (−0.727 − 0.686i)5-s + (−0.448 − 0.893i)6-s + (0.973 + 0.230i)7-s + (0.866 + 0.5i)8-s + (−0.286 + 0.957i)9-s + (−0.597 − 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.286 − 0.957i)12-s + (0.230 − 0.973i)13-s + (0.918 + 0.396i)14-s + (−0.116 + 0.993i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.687604379 - 0.8072215903i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.687604379 - 0.8072215903i\) |
\(L(1)\) |
\(\approx\) |
\(1.838605344 - 0.3496792095i\) |
\(L(1)\) |
\(\approx\) |
\(1.838605344 - 0.3496792095i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 3 | \( 1 + (-0.597 - 0.802i)T \) |
| 5 | \( 1 + (-0.727 - 0.686i)T \) |
| 7 | \( 1 + (0.973 + 0.230i)T \) |
| 11 | \( 1 + (0.597 - 0.802i)T \) |
| 13 | \( 1 + (0.230 - 0.973i)T \) |
| 17 | \( 1 + (0.642 - 0.766i)T \) |
| 19 | \( 1 + (0.342 + 0.939i)T \) |
| 23 | \( 1 + (-0.342 + 0.939i)T \) |
| 29 | \( 1 + (0.116 + 0.993i)T \) |
| 31 | \( 1 + (-0.230 - 0.973i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.984 + 0.173i)T \) |
| 47 | \( 1 + (0.0581 + 0.998i)T \) |
| 53 | \( 1 + (0.686 - 0.727i)T \) |
| 59 | \( 1 + (-0.116 + 0.993i)T \) |
| 61 | \( 1 + (0.998 + 0.0581i)T \) |
| 67 | \( 1 + (0.973 - 0.230i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + (-0.597 + 0.802i)T \) |
| 79 | \( 1 + (0.957 + 0.286i)T \) |
| 83 | \( 1 + (0.396 - 0.918i)T \) |
| 89 | \( 1 + (-0.448 - 0.893i)T \) |
| 97 | \( 1 + (0.957 - 0.286i)T \) |
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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
−18.27971390300425794890420164524, −17.55336287478511625814886636905, −16.756279164918699403029674707857, −16.22321599011727737992285704331, −15.25670475820289631556722314617, −15.069062219512145464378345732324, −14.296001613448284934414360423193, −13.91384892889792772706477224864, −12.595370738854170843729969635051, −11.93625748059925782424064162782, −11.5938155742546050845325778672, −10.897691342627719033909448337136, −10.4043185508282204732686222630, −9.61751426403793168246138955885, −8.55554074936545299619692014152, −7.640347672307740976222535913445, −6.77585308027781195410968212594, −6.438922118190496562834538911969, −5.355903202926053283675926210427, −4.6313362033656126657900984732, −4.12970942546056654696602795633, −3.647690930289355212293167857631, −2.58210795318246828066055257069, −1.66552508869228735591721505865, −0.64099213806784562005278868896,
0.87584961763324948473928106385, 1.257667324751449108730577088290, 2.30135152713287988467573087743, 3.367662751762732509367602447423, 3.99710473713943864524465593327, 5.05130170807380774884519040892, 5.497215851200579457418980368453, 5.94075308859512730272823724183, 7.11025039436535238689282593871, 7.7373150654364469499415529917, 8.08773312872987128729795416068, 8.935848819224208763674656491919, 10.32313644799779879497940976484, 11.178575037583538113375865175361, 11.62397866698199362957485037751, 12.06093103078746522917700971757, 12.74035890418765704840169091509, 13.3961381615528100508114337511, 14.19111426405796416597933847802, 14.60789179706525476366286950782, 15.732704083322279971958435058650, 16.07398483359423826737772624212, 16.888859108834773622040312579593, 17.35582764252386181690103524471, 18.26159313659250660410907377904