L(s) = 1 | + (−1.12 + 2.71i)5-s + (−3.03 + 3.03i)7-s + (−0.616 + 1.48i)11-s + (3.35 − 1.38i)13-s + 4.76·17-s + (−1.13 − 2.73i)19-s + (−4.11 + 4.11i)23-s + (−2.55 − 2.55i)25-s + (−8.16 + 3.38i)29-s − 6.16i·31-s + (−4.81 − 11.6i)35-s + (−9.38 − 3.88i)37-s + (−0.169 − 0.169i)41-s + (−7.57 − 3.13i)43-s + 2.44i·47-s + ⋯ |
L(s) = 1 | + (−0.502 + 1.21i)5-s + (−1.14 + 1.14i)7-s + (−0.185 + 0.449i)11-s + (0.929 − 0.385i)13-s + 1.15·17-s + (−0.259 − 0.627i)19-s + (−0.857 + 0.857i)23-s + (−0.511 − 0.511i)25-s + (−1.51 + 0.628i)29-s − 1.10i·31-s + (−0.814 − 1.96i)35-s + (−1.54 − 0.639i)37-s + (−0.0265 − 0.0265i)41-s + (−1.15 − 0.478i)43-s + 0.355i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5426188731\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5426188731\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.12 - 2.71i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (3.03 - 3.03i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.616 - 1.48i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-3.35 + 1.38i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 + (1.13 + 2.73i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (4.11 - 4.11i)T - 23iT^{2} \) |
| 29 | \( 1 + (8.16 - 3.38i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 6.16iT - 31T^{2} \) |
| 37 | \( 1 + (9.38 + 3.88i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (0.169 + 0.169i)T + 41iT^{2} \) |
| 43 | \( 1 + (7.57 + 3.13i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 2.44iT - 47T^{2} \) |
| 53 | \( 1 + (-1.99 - 0.824i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (1.21 + 0.503i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.04 - 2.52i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (3.91 - 1.62i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-5.28 - 5.28i)T + 71iT^{2} \) |
| 73 | \( 1 + (1.57 - 1.57i)T - 73iT^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + (3.31 - 1.37i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.99 + 6.99i)T - 89iT^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−10.17951202658702656022853882212, −9.512814183022408099901516656050, −8.643156806713661948298445371349, −7.62297933069157932603540150028, −6.94776873431757381962183173031, −6.01376475138062504722183684007, −5.43753491891606779030645177490, −3.67086843590937665434297882500, −3.26690337079042026861182885081, −2.12899075385328451707085135930,
0.23954798554574100887780348878, 1.44058931848170542818319018106, 3.47464402971789659996216613532, 3.85745748500002502270370892639, 4.98988593175418797878703396508, 6.02698664921086629717671263912, 6.82713041163253355137961431836, 7.906569623517951337300363723841, 8.447205405503355294331675008572, 9.364029653331602511091354703178