L(s) = 1 | − 3.69·5-s + (1.40 + 2.24i)7-s − 1.47·11-s + (−1.34 − 2.33i)13-s + (−3.28 − 5.69i)17-s + (0.444 − 0.769i)19-s + 6.28·23-s + 8.68·25-s + (−1.25 + 2.17i)29-s + (3.40 − 5.89i)31-s + (−5.19 − 8.29i)35-s + (−1.38 + 2.40i)37-s + (2.05 + 3.56i)41-s + (−0.00618 + 0.0107i)43-s + (3.49 + 6.05i)47-s + ⋯ |
L(s) = 1 | − 1.65·5-s + (0.531 + 0.847i)7-s − 0.445·11-s + (−0.374 − 0.648i)13-s + (−0.797 − 1.38i)17-s + (0.101 − 0.176i)19-s + 1.31·23-s + 1.73·25-s + (−0.233 + 0.403i)29-s + (0.611 − 1.05i)31-s + (−0.878 − 1.40i)35-s + (−0.228 + 0.395i)37-s + (0.321 + 0.556i)41-s + (−0.000943 + 0.00163i)43-s + (0.509 + 0.882i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9112168142\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9112168142\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.40 - 2.24i)T \) |
good | 5 | \( 1 + 3.69T + 5T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + (1.34 + 2.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.28 + 5.69i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.444 + 0.769i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 6.28T + 23T^{2} \) |
| 29 | \( 1 + (1.25 - 2.17i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.40 + 5.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.38 - 2.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.05 - 3.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.00618 - 0.0107i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.49 - 6.05i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.60 - 2.78i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.45 - 5.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.86 - 4.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.73 - 8.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + (6.03 + 10.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.72 - 9.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.23 + 3.87i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.43 + 7.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.58 - 11.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.839083785682893340318130139530, −7.967643201486467469165449611097, −7.54384488125181358179696970679, −6.84846782104924953839602657859, −5.65363887294831478649261758581, −4.83087433698599963435198240723, −4.37492768319048147274894619765, −3.07012230562984924755884628626, −2.60184900123411850407772288709, −0.832304183798444563607617541904,
0.40316216015751896798635060732, 1.75060424794875444397524649592, 3.13431535518149902949098713105, 4.00412831026464260914192353779, 4.45253581731526914383886076774, 5.28998434395987895784077409745, 6.66862331723558664414112443993, 7.12439388104526598059378564580, 7.87911650093288332425844197955, 8.396827055707767924454575005138