L(s) = 1 | + 2-s + 4-s + (1.15 − 1.99i)5-s + (2.61 − 0.396i)7-s + 8-s + (1.15 − 1.99i)10-s + (−2.92 + 5.06i)11-s + (3.58 + 0.349i)13-s + (2.61 − 0.396i)14-s + 16-s + 1.24·17-s + (2.09 + 3.63i)19-s + (1.15 − 1.99i)20-s + (−2.92 + 5.06i)22-s + 1.94·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.514 − 0.891i)5-s + (0.988 − 0.149i)7-s + 0.353·8-s + (0.364 − 0.630i)10-s + (−0.882 + 1.52i)11-s + (0.995 + 0.0968i)13-s + (0.699 − 0.105i)14-s + 0.250·16-s + 0.302·17-s + (0.481 + 0.833i)19-s + (0.257 − 0.445i)20-s + (−0.624 + 1.08i)22-s + 0.405·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.372541122\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.372541122\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.61 + 0.396i)T \) |
| 13 | \( 1 + (-3.58 - 0.349i)T \) |
good | 5 | \( 1 + (-1.15 + 1.99i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.92 - 5.06i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 1.24T + 17T^{2} \) |
| 19 | \( 1 + (-2.09 - 3.63i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.94T + 23T^{2} \) |
| 29 | \( 1 + (2.12 + 3.67i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.62 - 4.54i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 + (1.45 + 2.52i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.38 + 9.32i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.13 + 3.69i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.05 + 7.03i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 7.16T + 59T^{2} \) |
| 61 | \( 1 + (1.35 + 2.35i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.66 - 13.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.70 + 8.14i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.25 + 7.37i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.43 + 12.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.24T + 83T^{2} \) |
| 89 | \( 1 + 4.56T + 89T^{2} \) |
| 97 | \( 1 + (1.71 - 2.97i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.346852085390514448501344658203, −8.497050061832238605658313976409, −7.71173724691807062957214529971, −7.00573881442619196150857683052, −5.76681834979397081753554345802, −5.16921216566138923918077260635, −4.58447175455279024773780699814, −3.57319741907041565712559672195, −2.09765634428684888832234470850, −1.39493077092324424479514229490,
1.27264941424619306405037642684, 2.70854285237416141904819906246, 3.18455568034655642905722892782, 4.45633013720673105202986704956, 5.48849579828786160566316705465, 5.92535775552747407043991537873, 6.83596571527546122484709043442, 7.80667182102176005819105245340, 8.429721980541699913747972450839, 9.396833570620328318258107497783