L(s) = 1 | + (0.439 − 0.761i)2-s + (0.592 + 1.62i)3-s + (0.613 + 1.06i)4-s + (0.673 + 1.16i)5-s + (1.50 + 0.264i)6-s + 2.83·8-s + (−2.29 + 1.92i)9-s + 1.18·10-s + (−0.826 + 1.43i)11-s + (−1.36 + 1.62i)12-s + (−1.68 − 2.91i)13-s + (−1.5 + 1.78i)15-s + (0.0209 − 0.0362i)16-s − 0.467·17-s + (0.458 + 2.59i)18-s + 3.22·19-s + ⋯ |
L(s) = 1 | + (0.310 − 0.538i)2-s + (0.342 + 0.939i)3-s + (0.306 + 0.531i)4-s + (0.301 + 0.521i)5-s + (0.612 + 0.107i)6-s + 1.00·8-s + (−0.766 + 0.642i)9-s + 0.374·10-s + (−0.249 + 0.431i)11-s + (−0.394 + 0.469i)12-s + (−0.467 − 0.809i)13-s + (−0.387 + 0.461i)15-s + (0.00523 − 0.00906i)16-s − 0.113·17-s + (0.107 + 0.612i)18-s + 0.740·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75766 + 1.01478i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75766 + 1.01478i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.592 - 1.62i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.439 + 0.761i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.673 - 1.16i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.826 - 1.43i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.68 + 2.91i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 0.467T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 23 | \( 1 + (4.47 + 7.74i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.13 - 5.42i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.61 - 7.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + (-1.70 - 2.95i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.20 + 3.82i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.67 + 8.10i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.573T + 53T^{2} \) |
| 59 | \( 1 + (5.19 + 9.00i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.81 + 6.61i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.298 + 0.516i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 0.554T + 71T^{2} \) |
| 73 | \( 1 + 2.04T + 73T^{2} \) |
| 79 | \( 1 + (-1.20 + 2.08i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.52 - 13.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 + (0.949 - 1.64i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05903817618152992095877060906, −10.42711645245486462281823922585, −9.854613043510095830955265467381, −8.545489981828227928352176189639, −7.75055692631221141879414158576, −6.63756365350569371823518561563, −5.20956273626953559492910596511, −4.27156308938759156736074494962, −3.09027669157189502421267644268, −2.39409908794547250465066989338,
1.25276497863972632445077942768, 2.48036989865594839413021645218, 4.26779620305591964594936952992, 5.66869753929676337997398512140, 6.08326337723307081809231840340, 7.39206281579855011543294069398, 7.80310984625730523904742488893, 9.214368916876326436177218124052, 9.790939495778211645390508060313, 11.31185257848052783515927405108