L(s) = 1 | + (0.791 + 0.610i)2-s + (−0.951 + 0.309i)3-s + (0.254 + 0.967i)4-s + (−0.941 − 0.336i)6-s + (−0.389 + 0.921i)8-s + (0.809 − 0.587i)9-s + (−0.540 − 0.841i)12-s + (−0.931 − 0.362i)13-s + (−0.870 + 0.491i)16-s + (−0.717 − 0.696i)17-s + (0.999 + 0.0285i)18-s + (0.897 − 0.441i)19-s + (−0.909 − 0.415i)23-s + (0.0855 − 0.996i)24-s + (−0.516 − 0.856i)26-s + (−0.587 + 0.809i)27-s + ⋯ |
L(s) = 1 | + (0.791 + 0.610i)2-s + (−0.951 + 0.309i)3-s + (0.254 + 0.967i)4-s + (−0.941 − 0.336i)6-s + (−0.389 + 0.921i)8-s + (0.809 − 0.587i)9-s + (−0.540 − 0.841i)12-s + (−0.931 − 0.362i)13-s + (−0.870 + 0.491i)16-s + (−0.717 − 0.696i)17-s + (0.999 + 0.0285i)18-s + (0.897 − 0.441i)19-s + (−0.909 − 0.415i)23-s + (0.0855 − 0.996i)24-s + (−0.516 − 0.856i)26-s + (−0.587 + 0.809i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1610653308 + 0.7833255376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1610653308 + 0.7833255376i\) |
\(L(1)\) |
\(\approx\) |
\(0.8588169811 + 0.5394910884i\) |
\(L(1)\) |
\(\approx\) |
\(0.8588169811 + 0.5394910884i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.791 + 0.610i)T \) |
| 3 | \( 1 + (-0.951 + 0.309i)T \) |
| 13 | \( 1 + (-0.931 - 0.362i)T \) |
| 17 | \( 1 + (-0.717 - 0.696i)T \) |
| 19 | \( 1 + (0.897 - 0.441i)T \) |
| 23 | \( 1 + (-0.909 - 0.415i)T \) |
| 29 | \( 1 + (0.985 + 0.170i)T \) |
| 31 | \( 1 + (0.998 - 0.0570i)T \) |
| 37 | \( 1 + (0.633 + 0.774i)T \) |
| 41 | \( 1 + (0.564 + 0.825i)T \) |
| 43 | \( 1 + (0.755 + 0.654i)T \) |
| 47 | \( 1 + (-0.999 + 0.0285i)T \) |
| 53 | \( 1 + (-0.491 + 0.870i)T \) |
| 59 | \( 1 + (-0.564 + 0.825i)T \) |
| 61 | \( 1 + (-0.610 - 0.791i)T \) |
| 67 | \( 1 + (-0.281 - 0.959i)T \) |
| 71 | \( 1 + (-0.466 + 0.884i)T \) |
| 73 | \( 1 + (0.676 - 0.736i)T \) |
| 79 | \( 1 + (-0.974 - 0.226i)T \) |
| 83 | \( 1 + (0.980 + 0.198i)T \) |
| 89 | \( 1 + (-0.142 + 0.989i)T \) |
| 97 | \( 1 + (-0.996 - 0.0855i)T \) |
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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
−17.77994990659728918831882787767, −17.64346079423277798013808019554, −16.51210504560042401676117412885, −15.923416186725458643625430007505, −15.329232173141824612837794119841, −14.35779046570937108898206934462, −13.849691907493416961022565789275, −13.09134632365775011446522265750, −12.354525983089991822651366878417, −11.97932442520655362002937608118, −11.32998685522143419628819687848, −10.594700261566216492961276234787, −9.97722279163635390983171236863, −9.38165161983353233591025749746, −8.1274771471268078907734611979, −7.27915448386402747651122830761, −6.5781147158114351037485487659, −5.914239221907331300378907063003, −5.29363892237530013150280579154, −4.46712837247403860760466623921, −4.00302631175750338117575535214, −2.82305688097664788388644641314, −2.02989226217087380307174135227, −1.31509162080848956761149264533, −0.20106099976709779530603236485,
1.10618873871170031024023381085, 2.53259986310933797395517787857, 3.074489187698469107941832405536, 4.29826701024358601536191941966, 4.695548403649673019866272061437, 5.261619430823749728415094817411, 6.256742639403282987768250092657, 6.555874734043515748277351615347, 7.51468535431891950752983125408, 8.029950903862566474176421255857, 9.21271092771458790741989684254, 9.80018210923639361173000521078, 10.71329188594653743148136185773, 11.462451803583631173876721662485, 12.02609821774539312977941424537, 12.525810066761812606438441745924, 13.38163554714937304030828582028, 13.99597987129409408946519709894, 14.82160887307995295720992994553, 15.49335987148004204162488920728, 16.01375477783769315172779277580, 16.556447149347568835212411880570, 17.33730602762835236058806422836, 17.86920309918468404581224297457, 18.31279896132274633126803790385