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
−9.364029653331602511091354703178, −8.447205405503355294331675008572, −7.906569623517951337300363723841, −6.82713041163253355137961431836, −6.02698664921086629717671263912, −4.98988593175418797878703396508, −3.85745748500002502270370892639, −3.47464402971789659996216613532, −1.44058931848170542818319018106, −0.23954798554574100887780348878,
2.12899075385328451707085135930, 3.26690337079042026861182885081, 3.67086843590937665434297882500, 5.43753491891606779030645177490, 6.01376475138062504722183684007, 6.94776873431757381962183173031, 7.62297933069157932603540150028, 8.643156806713661948298445371349, 9.512814183022408099901516656050, 10.17951202658702656022853882212