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
−18.437431973668493093275603895858, −17.76166967780804906213761985695, −17.33399566748995686937296941610, −16.516425114397896323701830359408, −15.70529863108873281982200645349, −15.17043514221208492412985448504, −14.28982046974341561746005964223, −13.6073083814307335185832067223, −12.7020913667526691188896034788, −12.34638191978447748787679738213, −11.40976788557887954126202267729, −10.8376567867384747286210191523, −10.17934373182192740251309289094, −9.03016033957630277969788964747, −8.77823753055143796038157838486, −8.027840315495780257187190226, −7.348194144342144871055145432752, −6.70494526556882883153324139808, −6.03307079090479208159849901667, −5.31664612203245823083136416567, −3.769884839051375020905394651417, −3.210627842689612926326312256108, −2.25882535333264572768840336887, −1.56082139535162667040623236596, −0.860786630027220151649279629754,
0.44843847210910919205036800628, 1.627441130127545043675832841, 2.65467346182016987802880897358, 3.03971830428657693471591858320, 4.31092017856275396319593729993, 4.724364038870535486515640373906, 6.01510093040961006205826182751, 6.41578919925877348356470317942, 7.41884727005990098915407656033, 8.20232440840004702235526558651, 8.86024084243610293494860973116, 9.2960016499853808093056494936, 9.97523808352789332121174748382, 10.81999028210265450561200525849, 11.25333851724263107971376086011, 11.77106392565969619841121727347, 13.0565148046226800182370565776, 13.64776615402403730674653011788, 14.618565433585924859043870616120, 15.21195621861656752094841084228, 15.74505520727445571182656752983, 16.361828292633234548890383334586, 16.94243893182686028046358762041, 17.633746835572055345687398923846, 18.35234501065320933262414282185