L(s) = 1 | + (−0.0724 + 0.997i)2-s + (−0.906 + 0.421i)3-s + (−0.989 − 0.144i)4-s + (0.485 − 0.873i)5-s + (−0.354 − 0.935i)6-s + (0.262 + 0.964i)7-s + (0.215 − 0.976i)8-s + (0.644 − 0.764i)9-s + (0.836 + 0.548i)10-s + (0.958 − 0.285i)12-s + (0.998 + 0.0483i)13-s + (−0.981 + 0.192i)14-s + (−0.0724 + 0.997i)15-s + (0.958 + 0.285i)16-s + (0.568 − 0.822i)17-s + (0.715 + 0.698i)18-s + ⋯ |
L(s) = 1 | + (−0.0724 + 0.997i)2-s + (−0.906 + 0.421i)3-s + (−0.989 − 0.144i)4-s + (0.485 − 0.873i)5-s + (−0.354 − 0.935i)6-s + (0.262 + 0.964i)7-s + (0.215 − 0.976i)8-s + (0.644 − 0.764i)9-s + (0.836 + 0.548i)10-s + (0.958 − 0.285i)12-s + (0.998 + 0.0483i)13-s + (−0.981 + 0.192i)14-s + (−0.0724 + 0.997i)15-s + (0.958 + 0.285i)16-s + (0.568 − 0.822i)17-s + (0.715 + 0.698i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.082420524 + 0.04741885793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082420524 + 0.04741885793i\) |
\(L(1)\) |
\(\approx\) |
\(0.8014549732 + 0.2836895028i\) |
\(L(1)\) |
\(\approx\) |
\(0.8014549732 + 0.2836895028i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.0724 + 0.997i)T \) |
| 3 | \( 1 + (-0.906 + 0.421i)T \) |
| 5 | \( 1 + (0.485 - 0.873i)T \) |
| 7 | \( 1 + (0.262 + 0.964i)T \) |
| 13 | \( 1 + (0.998 + 0.0483i)T \) |
| 17 | \( 1 + (0.568 - 0.822i)T \) |
| 19 | \( 1 + (0.958 - 0.285i)T \) |
| 23 | \( 1 + (-0.215 - 0.976i)T \) |
| 29 | \( 1 + (0.926 - 0.377i)T \) |
| 31 | \( 1 + (-0.215 + 0.976i)T \) |
| 37 | \( 1 + (0.443 - 0.896i)T \) |
| 41 | \( 1 + (-0.568 - 0.822i)T \) |
| 43 | \( 1 + (-0.981 - 0.192i)T \) |
| 47 | \( 1 + (0.0724 - 0.997i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.943 + 0.331i)T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + (-0.981 + 0.192i)T \) |
| 71 | \( 1 + (-0.215 + 0.976i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.485 - 0.873i)T \) |
| 83 | \( 1 + (-0.443 - 0.896i)T \) |
| 89 | \( 1 + (0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.262 + 0.964i)T \) |
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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
−20.798344508628375934706719046013, −19.90719210588112688869171812205, −19.134748107459126396538042438692, −18.22557678744274396994544406337, −18.0978857877252698761560247748, −17.113183983204506505459217280703, −16.647298089714927131100836235741, −15.36903293767680787857440628661, −14.2573476606039216239083001280, −13.59052775683550120964487604298, −13.20048736699250049561569469733, −12.0737699580934583889905600486, −11.38577828870376348786501960800, −10.784849809131560289998278529707, −10.20310694118159072161351482295, −9.54929719836585999598317834575, −8.06794901913112328702217224230, −7.55044226899566566043963086218, −6.392281297577188775921866431072, −5.741294072277451879379429926344, −4.72645853623251815740826268728, −3.747933732847630450281258766328, −2.980648792062493858574724654432, −1.518203426649739310616017136341, −1.26042356849255907229023187758,
0.56474713623289677691039205181, 1.576162296065471104820711814850, 3.2658955721117960451955509424, 4.48276291468159157694212292570, 5.07861419431810281251564796835, 5.70039785077649102522609690844, 6.29855471554926246564027833032, 7.29972636786275009634959498395, 8.51254293431207566003698169078, 8.93403321471518967443161686010, 9.76149592711573010394027542986, 10.51110016150327369503194877974, 11.81078801049867591461449297225, 12.236989093718210797602255569107, 13.20990142491407959776958476092, 13.961118997862013246491312206077, 14.87331106358296957909494965164, 15.91317457462805932871340087340, 16.05272236816982867490924306528, 16.7931890779296354677966718836, 17.76487091679977510569142043669, 18.12604073885650572190110914925, 18.74489522752289631959309349201, 20.12450764975159239301818739001, 21.05524355340516473111810942665