L(s) = 1 | + (−1 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (−1.86 − 4.63i)5-s − 2.44·6-s + (2.71 − 2.71i)7-s + (2 + 2i)8-s + 2.99i·9-s + (6.50 + 2.77i)10-s − 20.1·11-s + (2.44 − 2.44i)12-s + (0.746 + 0.746i)13-s + 5.43i·14-s + (3.39 − 7.96i)15-s − 4·16-s + (0.556 − 0.556i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.373 − 0.927i)5-s − 0.408·6-s + (0.387 − 0.387i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.650 + 0.277i)10-s − 1.83·11-s + (0.204 − 0.204i)12-s + (0.0574 + 0.0574i)13-s + 0.387i·14-s + (0.226 − 0.531i)15-s − 0.250·16-s + (0.0327 − 0.0327i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8348574745\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8348574745\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 + (1.86 + 4.63i)T \) |
| 23 | \( 1 + (3.39 + 3.39i)T \) |
good | 7 | \( 1 + (-2.71 + 2.71i)T - 49iT^{2} \) |
| 11 | \( 1 + 20.1T + 121T^{2} \) |
| 13 | \( 1 + (-0.746 - 0.746i)T + 169iT^{2} \) |
| 17 | \( 1 + (-0.556 + 0.556i)T - 289iT^{2} \) |
| 19 | \( 1 - 19.9iT - 361T^{2} \) |
| 29 | \( 1 - 48.8iT - 841T^{2} \) |
| 31 | \( 1 - 50.2T + 961T^{2} \) |
| 37 | \( 1 + (16.5 - 16.5i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 57.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-58.6 - 58.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (41.3 - 41.3i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-28.0 - 28.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 46.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 80.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-3.70 + 3.70i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 45.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (19.9 + 19.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 11.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-51.9 - 51.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 153. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-109. + 109. i)T - 9.40e3iT^{2} \) |
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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
−10.44520472031456899017308986340, −9.607174679393298767384207218070, −8.686443199390544288639058804627, −7.895264571442833469999736462233, −7.62178652422675861222598454576, −6.02346646281875646677752721064, −5.03313355526629999968447871539, −4.38575125016375627089936164912, −2.88316939940725300051917292780, −1.27689795554124951882654943392,
0.35005137329993395073595834587, 2.37609537367635102066599728560, 2.72718017639550112206012560918, 4.12802097995058319611197245614, 5.46532118720835466728786363004, 6.68184471750213383848967710280, 7.72472592543574992793706882747, 8.020700435886190695912844593836, 9.074264654082219212719437348510, 10.16650040424487586321194429399