L(s) = 1 | + (−0.961 + 0.275i)2-s + (0.848 − 0.529i)4-s + (−0.766 − 0.642i)5-s + (−0.615 + 0.788i)7-s + (−0.669 + 0.743i)8-s + (0.913 + 0.406i)10-s + (0.615 − 0.788i)11-s + (−0.719 + 0.694i)13-s + (0.374 − 0.927i)14-s + (0.438 − 0.898i)16-s + (−0.309 − 0.951i)17-s + (−0.104 − 0.994i)19-s + (−0.990 − 0.139i)20-s + (−0.374 + 0.927i)22-s + (0.374 − 0.927i)23-s + ⋯ |
L(s) = 1 | + (−0.961 + 0.275i)2-s + (0.848 − 0.529i)4-s + (−0.766 − 0.642i)5-s + (−0.615 + 0.788i)7-s + (−0.669 + 0.743i)8-s + (0.913 + 0.406i)10-s + (0.615 − 0.788i)11-s + (−0.719 + 0.694i)13-s + (0.374 − 0.927i)14-s + (0.438 − 0.898i)16-s + (−0.309 − 0.951i)17-s + (−0.104 − 0.994i)19-s + (−0.990 − 0.139i)20-s + (−0.374 + 0.927i)22-s + (0.374 − 0.927i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.002453773170 + 0.007096569792i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.002453773170 + 0.007096569792i\) |
\(L(1)\) |
\(\approx\) |
\(0.5042765478 - 0.03651333916i\) |
\(L(1)\) |
\(\approx\) |
\(0.5042765478 - 0.03651333916i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.961 + 0.275i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.615 + 0.788i)T \) |
| 11 | \( 1 + (0.615 - 0.788i)T \) |
| 13 | \( 1 + (-0.719 + 0.694i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (0.374 - 0.927i)T \) |
| 29 | \( 1 + (-0.961 + 0.275i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.438 - 0.898i)T \) |
| 43 | \( 1 + (-0.241 + 0.970i)T \) |
| 47 | \( 1 + (0.997 - 0.0697i)T \) |
| 53 | \( 1 + (0.978 + 0.207i)T \) |
| 59 | \( 1 + (-0.961 - 0.275i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.669 + 0.743i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.882 + 0.469i)T \) |
| 83 | \( 1 + (0.241 - 0.970i)T \) |
| 89 | \( 1 + (0.978 - 0.207i)T \) |
| 97 | \( 1 + (0.990 + 0.139i)T \) |
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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
−21.63894575783983507337772156356, −20.31092659977086906988800523010, −19.991105537936148524447690667674, −19.27059405702027823720610739922, −18.61188165127519192574046989113, −17.52814413198975468438853286368, −17.024135678894903416471680416263, −16.18594767003831943358249123333, −15.11781086134947907085292982566, −14.807825983216520278702472463313, −13.28510378782641676490305351540, −12.43041694310854942165146903008, −11.71580014493243332865700190688, −10.739956543928749652058511738207, −10.14119867475936891988157012667, −9.43359313720337691999644276936, −8.205446154593716769619391184782, −7.44497659792142577338014445787, −6.935072211459199845812964850704, −5.918016218015437373342650637744, −4.118969985401880684405616169821, −3.57610459098841698008537545446, −2.48251444450085109126493620323, −1.266322259319847947281909637453, −0.00331340970288218055616437394,
0.772453030623873901997581089438, 2.210188418325968610201174464327, 3.1535916616794676084243155611, 4.54204233007443559900065796189, 5.5165400589605155294441142808, 6.591759438370374416025988574376, 7.2509819205490986853977854084, 8.354535043092570540581399007391, 9.198679395630421588298283139094, 9.33402301402689326774914295463, 10.85105391641757193761803997602, 11.61798643276650873106879253392, 12.163568231357295630538039306725, 13.24850480016743345455692930359, 14.495451985735465133768464200087, 15.2607320456090836589389733263, 16.06919498712888358534180863606, 16.570742447909123477398086759629, 17.27929050301108105113715777111, 18.56837502625010752379931483551, 18.917106359325941994803948984400, 19.784641792681944181897781179382, 20.25945771187691938706718085345, 21.40728159050910014047275970450, 22.197687304096057797538058218579