L(s) = 1 | + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.415 − 0.909i)10-s + (−0.959 − 0.281i)11-s + (0.415 + 0.909i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (0.841 + 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.654 − 0.755i)20-s − 22-s + (−0.959 + 0.281i)25-s + (0.654 + 0.755i)26-s + ⋯ |
L(s) = 1 | + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.415 − 0.909i)10-s + (−0.959 − 0.281i)11-s + (0.415 + 0.909i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (0.841 + 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.654 − 0.755i)20-s − 22-s + (−0.959 + 0.281i)25-s + (0.654 + 0.755i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.377938426 - 0.5032153484i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.377938426 - 0.5032153484i\) |
\(L(1)\) |
\(\approx\) |
\(1.497923106 - 0.3812671157i\) |
\(L(1)\) |
\(\approx\) |
\(1.497923106 - 0.3812671157i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 11 | \( 1 + (-0.959 - 0.281i)T \) |
| 13 | \( 1 + (0.415 + 0.909i)T \) |
| 17 | \( 1 + (0.841 + 0.540i)T \) |
| 19 | \( 1 + (-0.841 + 0.540i)T \) |
| 29 | \( 1 + (-0.841 - 0.540i)T \) |
| 31 | \( 1 + (-0.654 + 0.755i)T \) |
| 37 | \( 1 + (0.142 - 0.989i)T \) |
| 41 | \( 1 + (0.142 + 0.989i)T \) |
| 43 | \( 1 + (0.654 + 0.755i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.415 - 0.909i)T \) |
| 61 | \( 1 + (0.654 - 0.755i)T \) |
| 67 | \( 1 + (0.959 - 0.281i)T \) |
| 71 | \( 1 + (0.959 - 0.281i)T \) |
| 73 | \( 1 + (0.841 - 0.540i)T \) |
| 79 | \( 1 + (-0.415 - 0.909i)T \) |
| 83 | \( 1 + (-0.142 + 0.989i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (0.142 + 0.989i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−32.01812046447165901881754587151, −30.86690972492260914619378385967, −29.95002660786973928454729967565, −29.27293076044455545941056800932, −27.48187362104359267544923620962, −25.995485362609902668786867452330, −25.688036726869434397901435797633, −23.905098924510290528678325377300, −23.086939674898868441637212100091, −22.37116540974269782030749939326, −20.9884228055221157183924248658, −20.00469212885236328962554433005, −18.538916294070923517380226614097, −17.1210036629957997336706135248, −15.81597843994239708088872520234, −14.88116869887037289439033668173, −13.678640545564170070478381393188, −12.74437814802936570016273337101, −11.10284672644128410251800238965, −10.26655540541446648592102953668, −7.80547780273660577088821574584, −6.94654265525579188393662983138, −5.54435445472554231933230711262, −3.85535792105064962780380100994, −2.71256253902432592314717151711,
1.95354316575904223609827617475, 3.68429268297079273548229110118, 5.15760963915594365849335081010, 6.16601445759224821934210875586, 8.11143060697423245535834361267, 9.59066027977716646664127399093, 11.18207390114359311505688491320, 12.44748637509047669972940995232, 13.06437915589755892269146201137, 14.56358905329548413198528022595, 15.84034677666190876619865632287, 16.57158297290392375106986898013, 18.64955539727148244852119121247, 19.59203223771486213038273464965, 21.09950690773700674539283330027, 21.40609450997760720341155527507, 23.0528459169545313067407587704, 23.86788724597431353381957395310, 24.90209294308098071488996444712, 25.93791506699985795316959960291, 27.91962018142597418651897304930, 28.57243161795002295421744670136, 29.50661327860750454316731493317, 31.01278158255213694336585770935, 31.73391869666421496660240286791