| L(s) = 1 | + (0.368 + 1.69i)3-s + (0.355 − 0.978i)5-s + (0.272 − 1.54i)7-s + (−2.72 + 1.24i)9-s + (−1.44 − 3.97i)11-s + (−4.33 − 5.17i)13-s + (1.78 + 0.242i)15-s + (−0.494 − 0.857i)17-s + (−2.70 − 1.55i)19-s + (2.71 − 0.107i)21-s + (−1.17 − 6.68i)23-s + (3.00 + 2.51i)25-s + (−3.11 − 4.15i)27-s + (−2.84 + 3.39i)29-s + (0.409 + 2.32i)31-s + ⋯ |
| L(s) = 1 | + (0.212 + 0.977i)3-s + (0.159 − 0.437i)5-s + (0.102 − 0.583i)7-s + (−0.909 + 0.415i)9-s + (−0.436 − 1.19i)11-s + (−1.20 − 1.43i)13-s + (0.461 + 0.0625i)15-s + (−0.120 − 0.207i)17-s + (−0.619 − 0.357i)19-s + (0.591 − 0.0235i)21-s + (−0.245 − 1.39i)23-s + (0.600 + 0.503i)25-s + (−0.599 − 0.800i)27-s + (−0.529 + 0.630i)29-s + (0.0734 + 0.416i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0616 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0616 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.744582 - 0.699998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.744582 - 0.699998i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.368 - 1.69i)T \) |
| good | 5 | \( 1 + (-0.355 + 0.978i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.272 + 1.54i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (1.44 + 3.97i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (4.33 + 5.17i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.494 + 0.857i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.70 + 1.55i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.17 + 6.68i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.84 - 3.39i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.409 - 2.32i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (3.19 - 1.84i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.44 - 2.04i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.71 - 10.1i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.155 + 0.880i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 5.00iT - 53T^{2} \) |
| 59 | \( 1 + (-3.85 + 10.5i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (5.41 + 0.954i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.467 - 0.556i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.52 - 4.36i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.01 + 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.78 - 5.68i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.03 + 4.80i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-4.02 + 6.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.1 - 3.68i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.11435865469841871004514905172, −9.108232215032742482459198096609, −8.382601927628061656741244755855, −7.67851138345589798815076175928, −6.35853561140307131010925777489, −5.20439271282338953911035713194, −4.79054440035528729548350278810, −3.46846032038864915085332549374, −2.62630033340947302815364965378, −0.44591065389739924488654285835,
1.99796684244684306636462235400, 2.37011707948105714788681611698, 3.99444461453589020120416929403, 5.20400112278854198589769606089, 6.19174056723795907638448683852, 7.15160323888582831467552165025, 7.51005661975548822638979796578, 8.708677341227949962002600466430, 9.428729283462008921486770160470, 10.27318163951987963032820942384
