L(s) = 1 | + (0.895 − 1.55i)5-s + (−2.08 + 1.20i)7-s + (−1.36 + 0.790i)11-s + (−5.35 − 3.09i)13-s − 3.69i·17-s − 3.12·19-s + (1.36 − 2.35i)23-s + (0.896 + 1.55i)25-s + (−2.55 − 4.42i)29-s + (−5.95 − 3.43i)31-s + 4.31i·35-s + 5.24i·37-s + (5.32 + 3.07i)41-s + (0.452 + 0.783i)43-s + (−4.88 − 8.46i)47-s + ⋯ |
L(s) = 1 | + (0.400 − 0.693i)5-s + (−0.789 + 0.455i)7-s + (−0.412 + 0.238i)11-s + (−1.48 − 0.857i)13-s − 0.897i·17-s − 0.717·19-s + (0.283 − 0.491i)23-s + (0.179 + 0.310i)25-s + (−0.474 − 0.821i)29-s + (−1.06 − 0.617i)31-s + 0.729i·35-s + 0.861i·37-s + (0.831 + 0.479i)41-s + (0.0689 + 0.119i)43-s + (−0.713 − 1.23i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.142196 - 0.533315i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.142196 - 0.533315i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.895 + 1.55i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (2.08 - 1.20i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.36 - 0.790i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.35 + 3.09i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.69iT - 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + (-1.36 + 2.35i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.55 + 4.42i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.95 + 3.43i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.24iT - 37T^{2} \) |
| 41 | \( 1 + (-5.32 - 3.07i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.452 - 0.783i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.88 + 8.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 7.05T + 53T^{2} \) |
| 59 | \( 1 + (6.10 + 3.52i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.05 + 1.76i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.03 - 1.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.31T + 71T^{2} \) |
| 73 | \( 1 - 0.631T + 73T^{2} \) |
| 79 | \( 1 + (-7.82 + 4.51i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-13.5 + 7.82i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 1.16iT - 89T^{2} \) |
| 97 | \( 1 + (6.72 + 11.6i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.598427315453350200299967871849, −9.274575670613917083826630267574, −8.105970784718468011592141988760, −7.30830321565292506370881877085, −6.25212683287510446338046506622, −5.30312041604625851852117756862, −4.66618323585587472951400585717, −3.11289276034025910035594593362, −2.18949611156030473503409333196, −0.24216814536683917638334137122,
1.99250887143699407982993856228, 3.06374641609534827831803621882, 4.13181812737565870215753298618, 5.30371844550607343030894893183, 6.37752512466782798278567426427, 6.98926342049182032803218333277, 7.79628195670997927104879848894, 9.087221364411510493633073132530, 9.664038841111085105191366215542, 10.61915079921661510015397872644