L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (2.23 − 0.133i)5-s + 0.999·6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (1.99 + i)10-s + 4·11-s + (0.866 + 0.499i)12-s + (2.59 − 1.5i)13-s − 0.999·14-s + (1.86 − 1.23i)15-s + (−0.5 + 0.866i)16-s + (−5.19 − 3i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.998 − 0.0599i)5-s + 0.408·6-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.632 + 0.316i)10-s + 1.20·11-s + (0.249 + 0.144i)12-s + (0.720 − 0.416i)13-s − 0.267·14-s + (0.481 − 0.318i)15-s + (−0.125 + 0.216i)16-s + (−1.26 − 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.321671091\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.321671091\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-2.23 + 0.133i)T \) |
| 37 | \( 1 + (6.06 - 0.5i)T \) |
good | 7 | \( 1 + (0.866 - 0.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + (-2.59 + 1.5i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 41 | \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + (-3.46 - 2i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.92 + 4i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (14.7 + 8.5i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−9.568579287724510814297961603617, −9.076230820918730932770792876265, −8.325949989328414860830321789498, −7.05631788231048121868682064448, −6.53297331472064380813366707199, −5.77988465528278070469430885902, −4.72917163355928303756984101527, −3.62768767910899127005893585977, −2.65156489108211342245910221244, −1.48793541248492360513160215024,
1.51261396364571870057011780790, 2.44058524680606392032232969073, 3.71099829979146312959561252384, 4.30583963995840568401795585800, 5.51856044178916352168380622289, 6.49048557871274682785351128969, 6.86161109838808018166750636445, 8.558944044702380179922315792510, 9.023698738487869361382676493034, 9.840064793648448531732818624111