L(s) = 1 | + (0.241 − 0.970i)2-s + (−0.882 − 0.469i)4-s + (−0.766 − 0.642i)5-s + (−0.615 + 0.788i)7-s + (−0.669 + 0.743i)8-s + (−0.809 + 0.587i)10-s + (−0.990 − 0.139i)11-s + (−0.719 + 0.694i)13-s + (0.615 + 0.788i)14-s + (0.559 + 0.829i)16-s + (−0.669 + 0.743i)17-s + (−0.809 + 0.587i)19-s + (0.374 + 0.927i)20-s + (−0.374 + 0.927i)22-s + (−0.990 + 0.139i)23-s + ⋯ |
L(s) = 1 | + (0.241 − 0.970i)2-s + (−0.882 − 0.469i)4-s + (−0.766 − 0.642i)5-s + (−0.615 + 0.788i)7-s + (−0.669 + 0.743i)8-s + (−0.809 + 0.587i)10-s + (−0.990 − 0.139i)11-s + (−0.719 + 0.694i)13-s + (0.615 + 0.788i)14-s + (0.559 + 0.829i)16-s + (−0.669 + 0.743i)17-s + (−0.809 + 0.587i)19-s + (0.374 + 0.927i)20-s + (−0.374 + 0.927i)22-s + (−0.990 + 0.139i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1387163451 - 0.1814533377i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1387163451 - 0.1814533377i\) |
\(L(1)\) |
\(\approx\) |
\(0.5191099077 - 0.2291382543i\) |
\(L(1)\) |
\(\approx\) |
\(0.5191099077 - 0.2291382543i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.241 - 0.970i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.615 + 0.788i)T \) |
| 11 | \( 1 + (-0.990 - 0.139i)T \) |
| 13 | \( 1 + (-0.719 + 0.694i)T \) |
| 17 | \( 1 + (-0.669 + 0.743i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (-0.990 + 0.139i)T \) |
| 29 | \( 1 + (0.241 - 0.970i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.438 - 0.898i)T \) |
| 43 | \( 1 + (-0.241 + 0.970i)T \) |
| 47 | \( 1 + (-0.438 + 0.898i)T \) |
| 53 | \( 1 + (-0.669 + 0.743i)T \) |
| 59 | \( 1 + (0.719 - 0.694i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.669 + 0.743i)T \) |
| 79 | \( 1 + (-0.882 + 0.469i)T \) |
| 83 | \( 1 + (-0.961 + 0.275i)T \) |
| 89 | \( 1 + (0.978 - 0.207i)T \) |
| 97 | \( 1 + (-0.615 + 0.788i)T \) |
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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
−22.32429318122964298260717014762, −21.8405987073673115867135810427, −20.441562037616451367871151756189, −19.78641586090924413001982254007, −18.81756391982406354651779094199, −18.05888588090251460791290962228, −17.33161306509076358595725402910, −16.308327917569255770659898384576, −15.72780899782621222556183057839, −15.076445193417204863778674699, −14.225905066417500523609199295513, −13.31761160126178276917467445131, −12.7154627878781108169262766305, −11.67186102162374013130093797757, −10.45564581726298277812949376240, −9.946124560049984496802649318918, −8.59407333435707307205011543774, −7.835285397097392323035619958051, −7.02763178181799952001937217821, −6.57169902164774729624131423683, −5.21062870222921697974979703651, −4.44953146644809099578751465570, −3.4411768485943123700398909857, −2.633371681816185498920964572552, −0.311205439077772727953435584529,
0.141578364597019484027161619009, 1.80257548523456608516303770987, 2.58229276745372651419955233506, 3.756876931957298157303707236026, 4.4778028040147351510911249651, 5.43739701683218102333094273049, 6.33878870088853648404168249926, 7.87422290611905759544925569831, 8.56965363049224379060511364837, 9.40258353562467612839941086151, 10.24975287227484070078428589555, 11.20515486920656992254588502720, 12.07938404256598935539804586624, 12.59075349205311478764889121311, 13.2358374711876004827891271547, 14.34052824316088035513919463291, 15.30569886508057050692043893277, 15.90665684977360248985003600736, 16.95964025902715147685458779428, 17.92295056142754562368961045126, 18.96997882818636597792746696806, 19.27899897487179246277887447241, 20.03544677234660045290227343788, 21.04803924317118346747243004383, 21.513498926712266223134254099017