| L(s) = 1 | + (−0.0140 + 0.160i)2-s + (0.850 + 1.50i)3-s + (1.94 + 0.342i)4-s + (−1.03 − 1.97i)5-s + (−0.253 + 0.115i)6-s + (1.66 − 2.37i)7-s + (−0.165 + 0.617i)8-s + (−1.55 + 2.56i)9-s + (0.331 − 0.138i)10-s + (−1.48 + 4.09i)11-s + (1.13 + 3.22i)12-s + (−4.71 + 0.412i)13-s + (0.356 + 0.299i)14-s + (2.10 − 3.25i)15-s + (3.61 + 1.31i)16-s + (−1.51 − 5.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.00991 + 0.113i)2-s + (0.490 + 0.871i)3-s + (0.972 + 0.171i)4-s + (−0.464 − 0.885i)5-s + (−0.103 + 0.0469i)6-s + (0.627 − 0.896i)7-s + (−0.0585 + 0.218i)8-s + (−0.518 + 0.855i)9-s + (0.104 − 0.0439i)10-s + (−0.449 + 1.23i)11-s + (0.327 + 0.931i)12-s + (−1.30 + 0.114i)13-s + (0.0953 + 0.0800i)14-s + (0.543 − 0.839i)15-s + (0.903 + 0.328i)16-s + (−0.368 − 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31778 + 0.364872i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31778 + 0.364872i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.850 - 1.50i)T \) |
| 5 | \( 1 + (1.03 + 1.97i)T \) |
| good | 2 | \( 1 + (0.0140 - 0.160i)T + (-1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (-1.66 + 2.37i)T + (-2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (1.48 - 4.09i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (4.71 - 0.412i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (1.51 + 5.66i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.695 - 0.401i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.18 + 1.52i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (2.06 - 1.73i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.27 + 7.23i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.14 + 0.841i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.88 + 3.44i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.22 - 2.90i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-4.52 - 3.17i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-1.70 - 1.70i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.31 + 1.57i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.10 - 11.9i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.582 + 6.65i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (5.61 + 3.24i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (9.31 + 2.49i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.322 - 0.384i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.92 - 0.780i)T + (81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (-6.84 - 11.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.08 - 6.60i)T + (-62.3 + 74.3i)T^{2} \) |
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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
−13.34162063892871922729773422430, −12.15306418220581216401036531578, −11.27081548440403363228074670525, −10.21269600131083246299359273519, −9.230722847450989756619269752901, −7.74055524485548324372204414602, −7.34793418475231377570909423658, −5.07599302823867345245889975881, −4.30375230715336618387645502058, −2.41867206834719586693868758207,
2.17536824360686825787497637113, 3.18218733692087830546239452310, 5.69977664334104556063666451537, 6.73527812314529454391026836134, 7.76565636446138110373306315146, 8.594870075798726790353299638208, 10.32379625962508063237556300026, 11.34606642100942903111455753627, 11.95984711686605414489284719726, 13.00948686211596681881725772585
