L(s) = 1 | + (0.587 − 0.809i)2-s + (0.707 − 0.707i)3-s + (−0.309 − 0.951i)4-s + (0.951 − 0.309i)5-s + (−0.156 − 0.987i)6-s + (−0.156 + 0.987i)7-s + (−0.951 − 0.309i)8-s − i·9-s + (0.309 − 0.951i)10-s + (0.453 + 0.891i)11-s + (−0.891 − 0.453i)12-s + (−0.987 + 0.156i)13-s + (0.707 + 0.707i)14-s + (0.453 − 0.891i)15-s + (−0.809 + 0.587i)16-s + (0.891 − 0.453i)17-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)2-s + (0.707 − 0.707i)3-s + (−0.309 − 0.951i)4-s + (0.951 − 0.309i)5-s + (−0.156 − 0.987i)6-s + (−0.156 + 0.987i)7-s + (−0.951 − 0.309i)8-s − i·9-s + (0.309 − 0.951i)10-s + (0.453 + 0.891i)11-s + (−0.891 − 0.453i)12-s + (−0.987 + 0.156i)13-s + (0.707 + 0.707i)14-s + (0.453 − 0.891i)15-s + (−0.809 + 0.587i)16-s + (0.891 − 0.453i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.570716065 - 1.982039782i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.570716065 - 1.982039782i\) |
\(L(1)\) |
\(\approx\) |
\(1.452804593 - 1.114671296i\) |
\(L(1)\) |
\(\approx\) |
\(1.452804593 - 1.114671296i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.951 - 0.309i)T \) |
| 7 | \( 1 + (-0.156 + 0.987i)T \) |
| 11 | \( 1 + (0.453 + 0.891i)T \) |
| 13 | \( 1 + (-0.987 + 0.156i)T \) |
| 17 | \( 1 + (0.891 - 0.453i)T \) |
| 19 | \( 1 + (-0.987 - 0.156i)T \) |
| 23 | \( 1 + (0.809 + 0.587i)T \) |
| 29 | \( 1 + (0.891 + 0.453i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + (0.587 - 0.809i)T \) |
| 47 | \( 1 + (-0.156 - 0.987i)T \) |
| 53 | \( 1 + (-0.891 - 0.453i)T \) |
| 59 | \( 1 + (-0.809 - 0.587i)T \) |
| 61 | \( 1 + (-0.587 - 0.809i)T \) |
| 67 | \( 1 + (-0.453 + 0.891i)T \) |
| 71 | \( 1 + (-0.453 - 0.891i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (-0.707 + 0.707i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.156 + 0.987i)T \) |
| 97 | \( 1 + (0.453 - 0.891i)T \) |
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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
−34.57098790426990201141850914392, −33.522633394099052618542179210216, −32.58469537248153062875023733837, −32.000233759695170456693017947283, −30.377778519836176464964324349162, −29.54096698115388632515263158178, −27.27377209907068440813909275795, −26.43580576020041911315940732984, −25.48558984512545688404173237536, −24.45647919862998073313048741708, −22.839112048069098682824138749379, −21.710317605989105955558236557544, −20.910111691049616261153033230501, −19.241667951987845230366258909078, −17.199880140666461416988891797784, −16.53156006735854602328917409473, −14.76518844354503528328120161563, −14.13073013406655189309846055288, −12.98041660156207099618319267555, −10.6118892983353565584003562852, −9.25479772579750361960605368941, −7.7087857650828650168384003102, −6.10265265287133442535077538189, −4.436559627804492888405206485303, −2.95909149414783302258387143758,
1.66914631617698378760206868839, 2.817653428584656925236350726303, 5.06359892797314647829820109254, 6.63213244811130860440130627372, 8.95352462567187507425350069846, 9.85139962292055437059502507244, 12.1130185448413933660405472985, 12.76726550614957559548941324263, 14.15031711313357200026494310993, 15.05104321582336742796695361485, 17.50907556297056581254913560687, 18.712281595460906208190854616651, 19.77257706605023346748772617837, 21.00082064045830516776659731997, 21.94631223158596377210104728741, 23.47228349927807842680307967447, 24.87345652742755983481181457849, 25.456946076109048328831399745768, 27.51767935216950258284310375231, 28.85816157575666729425208724196, 29.61592123492115326208788431231, 30.79074445437229591162652202513, 31.805001037979794943903604709215, 32.629445376657949226789228676460, 34.197158557060623473772740328618