L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−1.25 + 1.72i)5-s + (−0.951 − 0.309i)7-s + (−0.309 − 0.951i)8-s + 2.12i·10-s + (2.72 − 1.88i)11-s + (0.138 + 0.190i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (3.83 + 2.78i)17-s + (2.37 − 0.770i)19-s + (1.25 + 1.72i)20-s + (1.09 − 3.13i)22-s + 7.44i·23-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.559 + 0.770i)5-s + (−0.359 − 0.116i)7-s + (−0.109 − 0.336i)8-s + 0.673i·10-s + (0.821 − 0.569i)11-s + (0.0383 + 0.0527i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (0.931 + 0.676i)17-s + (0.543 − 0.176i)19-s + (0.279 + 0.385i)20-s + (0.233 − 0.667i)22-s + 1.55i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.165783786\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.165783786\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.72 + 1.88i)T \) |
good | 5 | \( 1 + (1.25 - 1.72i)T + (-1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (-0.138 - 0.190i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.83 - 2.78i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.37 + 0.770i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 7.44iT - 23T^{2} \) |
| 29 | \( 1 + (-1.54 + 4.75i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.902 + 0.656i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.09 + 9.53i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.856 - 2.63i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 3.37iT - 43T^{2} \) |
| 47 | \( 1 + (-11.7 + 3.80i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.73 - 6.51i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-12.9 - 4.20i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (7.05 - 9.70i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 1.74T + 67T^{2} \) |
| 71 | \( 1 + (-5.46 + 7.51i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.11 - 1.33i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (4.17 + 5.74i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.89 - 1.37i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 9.21iT - 89T^{2} \) |
| 97 | \( 1 + (-0.932 + 0.677i)T + (29.9 - 92.2i)T^{2} \) |
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
−9.665896439184575780076903429097, −8.903521024344801978480198628013, −7.66972783772814902977964510743, −7.14012915993509430050260858415, −6.06714533566656156685444529885, −5.51173638708618309340288790556, −3.96455696252981410010105441499, −3.64898124632479933279257577815, −2.61062251571655182886722384748, −1.08362440693107715420591191493,
0.982440553752426554061191931509, 2.68422773328604600140942944836, 3.75421105632977921922628166636, 4.59467833324548923582685444910, 5.27868312309239675346140654983, 6.38706589695434599988773343772, 7.04489718490125524588713330845, 7.990718295432839877497009271750, 8.654612579249927802224881015900, 9.492719452553635603068999094690