L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.450 + 0.619i)5-s + (0.951 + 0.309i)7-s + (0.309 + 0.951i)8-s − 0.765i·10-s + (−2.23 − 2.45i)11-s + (2.73 + 3.76i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (−0.239 − 0.174i)17-s + (1.24 − 0.403i)19-s + (0.450 + 0.619i)20-s + (3.24 + 0.671i)22-s − 2.28i·23-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.201 + 0.276i)5-s + (0.359 + 0.116i)7-s + (0.109 + 0.336i)8-s − 0.242i·10-s + (−0.673 − 0.739i)11-s + (0.758 + 1.04i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (−0.0581 − 0.0422i)17-s + (0.284 − 0.0926i)19-s + (0.100 + 0.138i)20-s + (0.692 + 0.143i)22-s − 0.475i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.117549415\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.117549415\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.951 - 0.309i)T \) |
| 11 | \( 1 + (2.23 + 2.45i)T \) |
good | 5 | \( 1 + (0.450 - 0.619i)T + (-1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (-2.73 - 3.76i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.239 + 0.174i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.24 + 0.403i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 2.28iT - 23T^{2} \) |
| 29 | \( 1 + (1.67 - 5.16i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.72 + 1.98i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.636 + 1.95i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.20 - 6.79i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 0.901iT - 43T^{2} \) |
| 47 | \( 1 + (-1.58 + 0.516i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.589 - 0.811i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.26 - 1.38i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.44 - 1.98i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 7.33T + 67T^{2} \) |
| 71 | \( 1 + (7.11 - 9.78i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.0514 + 0.0167i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-3.07 - 4.23i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.85 - 4.97i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 + (-2.24 + 1.63i)T + (29.9 - 92.2i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.584874597033803107669151949002, −8.839877717642206004503401006045, −8.226995830569582579431070302744, −7.38286931450026387309371141771, −6.60326471170424055697466148602, −5.76358902686703029448866904802, −4.86684232430902521026629713704, −3.70984078137210965897471034270, −2.52765571815495541210961685595, −1.16830675218739514824514306809,
0.64243934779637956698270227472, 1.98369486916961101701479434685, 3.10445431458998412540505453397, 4.16984962527941723376663966217, 5.12205200311543595413318159098, 6.09622476091528954814609266302, 7.26750135168352100732395430880, 7.945583304999031169454688745799, 8.494280756787789232705945198792, 9.450017352825220930203413343531