L(s) = 1 | − 2-s + (0.893 + 0.448i)3-s + 4-s + (0.993 + 0.116i)5-s + (−0.893 − 0.448i)6-s + (0.597 − 0.802i)7-s − 8-s + (0.597 + 0.802i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (0.893 + 0.448i)12-s + (0.993 + 0.116i)13-s + (−0.597 + 0.802i)14-s + (0.835 + 0.549i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | − 2-s + (0.893 + 0.448i)3-s + 4-s + (0.993 + 0.116i)5-s + (−0.893 − 0.448i)6-s + (0.597 − 0.802i)7-s − 8-s + (0.597 + 0.802i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (0.893 + 0.448i)12-s + (0.993 + 0.116i)13-s + (−0.597 + 0.802i)14-s + (0.835 + 0.549i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.214868210 + 0.07785760468i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.214868210 + 0.07785760468i\) |
\(L(1)\) |
\(\approx\) |
\(1.248294394 + 0.09256556087i\) |
\(L(1)\) |
\(\approx\) |
\(1.248294394 + 0.09256556087i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.893 + 0.448i)T \) |
| 5 | \( 1 + (0.993 + 0.116i)T \) |
| 7 | \( 1 + (0.597 - 0.802i)T \) |
| 11 | \( 1 + (-0.993 + 0.116i)T \) |
| 13 | \( 1 + (0.993 + 0.116i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (-0.597 + 0.802i)T \) |
| 31 | \( 1 + (-0.973 - 0.230i)T \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 + (-0.173 - 0.984i)T \) |
| 47 | \( 1 + (0.893 - 0.448i)T \) |
| 53 | \( 1 + (-0.0581 - 0.998i)T \) |
| 59 | \( 1 + (0.835 + 0.549i)T \) |
| 61 | \( 1 + (0.286 - 0.957i)T \) |
| 67 | \( 1 + (0.893 - 0.448i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.597 + 0.802i)T \) |
| 79 | \( 1 + (0.835 + 0.549i)T \) |
| 83 | \( 1 + (-0.686 - 0.727i)T \) |
| 89 | \( 1 + (-0.396 - 0.918i)T \) |
| 97 | \( 1 + (-0.973 + 0.230i)T \) |
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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
−18.363482814093363983689650830193, −17.94778735538872673533117863303, −17.4760760522679354864569610484, −16.47499828476484081590094030196, −15.658664398261728314460153903188, −15.193721230923800337838127455056, −14.47072465424180011669147120696, −13.677290379640248871031060883158, −12.93091688639767417737890892313, −12.4675177998876982326478270049, −11.36616340768091000483822145882, −10.79581518930984793796154390595, −9.9364827975441033344723648430, −9.29543091887840694799342465223, −8.761563419073575316890555413786, −8.05333567955892756587158396458, −7.67183796476696781700340808704, −6.55802335592994295510588611185, −5.91257278219289898750522532253, −5.356099082916042033992370787568, −3.951419539496107923387159523920, −2.90436137425930922505548701745, −2.34621806331082509086648482830, −1.71411749703060231556649546397, −0.99607803382928454453778612923,
0.84235644074857061603881049553, 1.9196013073639032248356242140, 2.17668422677615310701622396532, 3.30418427084945947495413764712, 3.93135170323986396506594497751, 5.199291209454959142868842194405, 5.71499626321299044914135135479, 6.90124413320297814763582856055, 7.42146740625153109769154714912, 8.24228505567449678200581833938, 8.6523621357410313673959233371, 9.64104686953806709120840127200, 10.000578893895537037312239129669, 10.79040611444075613020006802519, 11.0050461752637485870438189104, 12.375359906760753107869733453804, 13.10939245833958802543324275491, 13.99295526209527858978712249571, 14.30099438087733982227099552221, 15.09973284385951332166491760861, 16.01867539382708187789990151558, 16.42429291451423544993382110432, 17.07029272207262222287769792165, 18.04016330316025071462690234983, 18.46740414718330939086105316061