| L(s) = 1 | + (−0.000656 + 1.73i)3-s + (1.14 − 3.14i)5-s + (−0.868 − 0.153i)7-s + (−2.99 − 0.00227i)9-s + (−2.12 + 0.773i)11-s + (2.22 − 1.86i)13-s + (5.44 + 1.98i)15-s + (3.66 − 2.11i)17-s + (2.73 + 1.57i)19-s + (0.265 − 1.50i)21-s + (−0.991 − 5.62i)23-s + (−4.74 − 3.98i)25-s + (0.00590 − 5.19i)27-s + (3.68 − 4.39i)29-s + (6.00 − 1.05i)31-s + ⋯ |
| L(s) = 1 | + (−0.000378 + 0.999i)3-s + (0.511 − 1.40i)5-s + (−0.328 − 0.0578i)7-s + (−0.999 − 0.000757i)9-s + (−0.641 + 0.233i)11-s + (0.617 − 0.518i)13-s + (1.40 + 0.512i)15-s + (0.889 − 0.513i)17-s + (0.626 + 0.361i)19-s + (0.0580 − 0.328i)21-s + (−0.206 − 1.17i)23-s + (−0.949 − 0.796i)25-s + (0.00113 − 0.999i)27-s + (0.685 − 0.816i)29-s + (1.07 − 0.190i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42016 - 0.494531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42016 - 0.494531i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.000656 - 1.73i)T \) |
| good | 5 | \( 1 + (-1.14 + 3.14i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.868 + 0.153i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.12 - 0.773i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.22 + 1.86i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.66 + 2.11i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.73 - 1.57i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.991 + 5.62i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.68 + 4.39i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.00 + 1.05i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.04 - 1.80i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.86 + 4.60i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.42 + 3.92i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.42 + 8.08i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 9.10iT - 53T^{2} \) |
| 59 | \( 1 + (-7.34 - 2.67i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.35 - 7.67i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.03 - 2.42i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (5.53 + 9.58i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.03 + 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.52 - 6.58i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-9.67 - 8.11i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-11.7 - 6.80i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.8 - 3.95i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.09144357806766437751548583637, −9.319136682435924163580864514226, −8.506982765473286633722132837103, −7.907904851105661626467000472746, −6.30001058946610766674506431469, −5.40516989392641682475424574672, −4.88137543715376909967746442735, −3.81979907616362637271897365207, −2.60951032257272892564472097685, −0.78183204438496672439988183998,
1.48151893683466814929671053097, 2.79231909913398813822146800332, 3.38352193981600238335976723008, 5.28619471579778420911948125127, 6.21058985606709109111207122488, 6.69436814717617237723705083314, 7.60220543567492826068842334222, 8.338682109157627534870419186553, 9.549381864919879048840135452256, 10.27687971524746730765072963554
