| L(s) = 1 | + (−0.882 + 1.10i)2-s + (−0.603 + 1.82i)3-s + (−0.441 − 1.95i)4-s + (0.147 − 0.382i)5-s + (−1.48 − 2.27i)6-s + (1.09 − 1.83i)7-s + (2.54 + 1.23i)8-s + (−0.559 − 0.415i)9-s + (0.292 + 0.500i)10-s + (0.0203 + 0.165i)11-s + (3.82 + 0.369i)12-s + (−0.156 + 6.37i)13-s + (1.05 + 2.83i)14-s + (0.609 + 0.499i)15-s + (−3.60 + 1.72i)16-s + (−2.24 − 2.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.624 + 0.781i)2-s + (−0.348 + 1.05i)3-s + (−0.220 − 0.975i)4-s + (0.0659 − 0.170i)5-s + (−0.606 − 0.929i)6-s + (0.415 − 0.692i)7-s + (0.899 + 0.436i)8-s + (−0.186 − 0.138i)9-s + (0.0924 + 0.158i)10-s + (0.00613 + 0.0497i)11-s + (1.10 + 0.106i)12-s + (−0.0434 + 1.76i)13-s + (0.282 + 0.756i)14-s + (0.157 + 0.129i)15-s + (−0.902 + 0.431i)16-s + (−0.545 − 0.664i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.247273 + 0.828066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.247273 + 0.828066i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.882 - 1.10i)T \) |
| good | 3 | \( 1 + (0.603 - 1.82i)T + (-2.40 - 1.78i)T^{2} \) |
| 5 | \( 1 + (-0.147 + 0.382i)T + (-3.70 - 3.35i)T^{2} \) |
| 7 | \( 1 + (-1.09 + 1.83i)T + (-3.29 - 6.17i)T^{2} \) |
| 11 | \( 1 + (-0.0203 - 0.165i)T + (-10.6 + 2.67i)T^{2} \) |
| 13 | \( 1 + (0.156 - 6.37i)T + (-12.9 - 0.637i)T^{2} \) |
| 17 | \( 1 + (2.24 + 2.73i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-3.95 - 2.78i)T + (6.40 + 17.8i)T^{2} \) |
| 23 | \( 1 + (2.39 - 5.05i)T + (-14.5 - 17.7i)T^{2} \) |
| 29 | \( 1 + (-7.03 + 1.94i)T + (24.8 - 14.9i)T^{2} \) |
| 31 | \( 1 + (9.03 - 1.79i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (0.286 + 1.27i)T + (-33.4 + 15.8i)T^{2} \) |
| 41 | \( 1 + (-6.61 + 5.99i)T + (4.01 - 40.8i)T^{2} \) |
| 43 | \( 1 + (2.75 - 5.46i)T + (-25.6 - 34.5i)T^{2} \) |
| 47 | \( 1 + (5.22 + 2.79i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (10.3 + 2.85i)T + (45.4 + 27.2i)T^{2} \) |
| 59 | \( 1 + (-7.83 + 0.192i)T + (58.9 - 2.89i)T^{2} \) |
| 61 | \( 1 + (-7.97 - 6.88i)T + (8.95 + 60.3i)T^{2} \) |
| 67 | \( 1 + (4.19 - 0.309i)T + (66.2 - 9.83i)T^{2} \) |
| 71 | \( 1 + (1.12 - 7.56i)T + (-67.9 - 20.6i)T^{2} \) |
| 73 | \( 1 + (-6.86 - 11.4i)T + (-34.4 + 64.3i)T^{2} \) |
| 79 | \( 1 + (12.0 + 3.64i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-0.162 + 0.725i)T + (-75.0 - 35.4i)T^{2} \) |
| 89 | \( 1 + (2.12 + 4.49i)T + (-56.4 + 68.7i)T^{2} \) |
| 97 | \( 1 + (4.21 + 2.81i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.14318845198707498332187135874, −10.09021154383066244692308664808, −9.517837008214248234544989085557, −8.806887687083063022761269244987, −7.51723157187865728866398763529, −6.89810446596806622779365645590, −5.57311757538808231916828760348, −4.74667911133026200603909723628, −3.97283671576490471308610664278, −1.58026253486278132644371816703,
0.70575309457715067094165939945, 2.10619719119721048036172802076, 3.15892582013776691503710597341, 4.80844288964970397402739739357, 6.05564965211838657585708738638, 7.07095273611205938529711647441, 8.024621599252222088327138626687, 8.581057833970398359364069994408, 9.780701009820926783917027055747, 10.68606007204252957807290753711
