L(s) = 1 | + (−0.998 − 0.0570i)2-s + (0.309 − 0.951i)3-s + (0.993 + 0.113i)4-s + (−0.362 + 0.931i)6-s + (−0.985 − 0.170i)8-s + (−0.809 − 0.587i)9-s + (0.415 − 0.909i)12-s + (0.870 − 0.491i)13-s + (0.974 + 0.226i)16-s + (−0.941 + 0.336i)17-s + (0.774 + 0.633i)18-s + (−0.0285 + 0.999i)19-s + (0.654 + 0.755i)23-s + (−0.466 + 0.884i)24-s + (−0.897 + 0.441i)26-s + (−0.809 + 0.587i)27-s + ⋯ |
L(s) = 1 | + (−0.998 − 0.0570i)2-s + (0.309 − 0.951i)3-s + (0.993 + 0.113i)4-s + (−0.362 + 0.931i)6-s + (−0.985 − 0.170i)8-s + (−0.809 − 0.587i)9-s + (0.415 − 0.909i)12-s + (0.870 − 0.491i)13-s + (0.974 + 0.226i)16-s + (−0.941 + 0.336i)17-s + (0.774 + 0.633i)18-s + (−0.0285 + 0.999i)19-s + (0.654 + 0.755i)23-s + (−0.466 + 0.884i)24-s + (−0.897 + 0.441i)26-s + (−0.809 + 0.587i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9989431018 + 0.03762517264i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9989431018 + 0.03762517264i\) |
\(L(1)\) |
\(\approx\) |
\(0.7315807606 - 0.1778819978i\) |
\(L(1)\) |
\(\approx\) |
\(0.7315807606 - 0.1778819978i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.998 - 0.0570i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 13 | \( 1 + (0.870 - 0.491i)T \) |
| 17 | \( 1 + (-0.941 + 0.336i)T \) |
| 19 | \( 1 + (-0.0285 + 0.999i)T \) |
| 23 | \( 1 + (0.654 + 0.755i)T \) |
| 29 | \( 1 + (0.564 + 0.825i)T \) |
| 31 | \( 1 + (-0.198 + 0.980i)T \) |
| 37 | \( 1 + (0.736 - 0.676i)T \) |
| 41 | \( 1 + (-0.254 - 0.967i)T \) |
| 43 | \( 1 + (-0.142 - 0.989i)T \) |
| 47 | \( 1 + (0.774 - 0.633i)T \) |
| 53 | \( 1 + (-0.974 + 0.226i)T \) |
| 59 | \( 1 + (0.254 - 0.967i)T \) |
| 61 | \( 1 + (-0.998 + 0.0570i)T \) |
| 67 | \( 1 + (-0.841 + 0.540i)T \) |
| 71 | \( 1 + (0.610 - 0.791i)T \) |
| 73 | \( 1 + (-0.516 + 0.856i)T \) |
| 79 | \( 1 + (-0.696 + 0.717i)T \) |
| 83 | \( 1 + (-0.0855 + 0.996i)T \) |
| 89 | \( 1 + (0.959 + 0.281i)T \) |
| 97 | \( 1 + (-0.466 + 0.884i)T \) |
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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
−18.35234501065320933262414282185, −17.633746835572055345687398923846, −16.94243893182686028046358762041, −16.361828292633234548890383334586, −15.74505520727445571182656752983, −15.21195621861656752094841084228, −14.618565433585924859043870616120, −13.64776615402403730674653011788, −13.0565148046226800182370565776, −11.77106392565969619841121727347, −11.25333851724263107971376086011, −10.81999028210265450561200525849, −9.97523808352789332121174748382, −9.2960016499853808093056494936, −8.86024084243610293494860973116, −8.20232440840004702235526558651, −7.41884727005990098915407656033, −6.41578919925877348356470317942, −6.01510093040961006205826182751, −4.724364038870535486515640373906, −4.31092017856275396319593729993, −3.03971830428657693471591858320, −2.65467346182016987802880897358, −1.627441130127545043675832841, −0.44843847210910919205036800628,
0.860786630027220151649279629754, 1.56082139535162667040623236596, 2.25882535333264572768840336887, 3.210627842689612926326312256108, 3.769884839051375020905394651417, 5.31664612203245823083136416567, 6.03307079090479208159849901667, 6.70494526556882883153324139808, 7.348194144342144871055145432752, 8.027840315495780257187190226, 8.77823753055143796038157838486, 9.03016033957630277969788964747, 10.17934373182192740251309289094, 10.8376567867384747286210191523, 11.40976788557887954126202267729, 12.34638191978447748787679738213, 12.7020913667526691188896034788, 13.6073083814307335185832067223, 14.28982046974341561746005964223, 15.17043514221208492412985448504, 15.70529863108873281982200645349, 16.516425114397896323701830359408, 17.33399566748995686937296941610, 17.76166967780804906213761985695, 18.437431973668493093275603895858