L(s) = 1 | − i·7-s − 11-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)17-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)29-s + 31-s − i·37-s + (−0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + (0.866 + 0.5i)47-s − 49-s + (0.866 + 0.5i)53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯ |
L(s) = 1 | − i·7-s − 11-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)17-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)29-s + 31-s − i·37-s + (−0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + (0.866 + 0.5i)47-s − 49-s + (0.866 + 0.5i)53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.308039290 + 0.04443595656i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.308039290 + 0.04443595656i\) |
\(L(1)\) |
\(\approx\) |
\(0.9719094975 + 0.05008466993i\) |
\(L(1)\) |
\(\approx\) |
\(0.9719094975 + 0.05008466993i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.866 - 0.5i)T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−19.52549413062999876231738790572, −19.09872297674133868135244297818, −18.17573729853347163988894502346, −17.41404250748897475576949334171, −16.803316292319115530460168174704, −16.18869145489908404703391606927, −15.2775254005614320779223244095, −14.61569314737158872619271258252, −13.72106395738310643064486261291, −13.32785438225863297209744782600, −12.335474777617050177636767340335, −11.69003738119312892716905070811, −10.78075908633678684710807428199, −9.96439824929114259073148063793, −9.73322200919210896856596499450, −8.31046560839306819753524219358, −7.80251604756775839499097910812, −7.11340082972431518204722195224, −6.19674063972066934607498663810, −5.3082152193397977322225097752, −4.484923557265376292277509091370, −3.74154678087756491648744845209, −2.76174422380882318593975069472, −1.84187604631614782847908862013, −0.70549342775337906440558303710,
0.64196330089746958015500267521, 2.15322538106452152006622281924, 2.615282342648416110698499686523, 3.554898491030004532887262954883, 4.750375894262622239337621629, 5.43220528737302131489135352821, 5.94979637616247361074636725761, 7.18180244845734582216730989068, 7.77387164449447381423300269229, 8.5780375495894692366443287611, 9.368030187141103888766032084508, 10.1783445463727872356682226423, 10.75489783992673408263816742985, 11.9250188154841165428638292892, 12.30359973483718785333973953635, 12.98589332767277073389538028099, 14.01301735821285817434057806255, 14.60701548813330968383824987118, 15.489436742655532058275325853757, 15.89597012113405186510791923406, 16.77520766498849598922442271994, 17.63384220999289905830554123011, 18.32914976456017835898681147979, 18.80528387289557052147806554263, 19.61074965129557401377865929761