L(s) = 1 | − 2-s + (−1 − 1.41i)3-s + 4-s − i·5-s + (1 + 1.41i)6-s + 0.585·7-s − 8-s + (−1.00 + 2.82i)9-s + i·10-s + 5.41i·11-s + (−1 − 1.41i)12-s + 2.24i·13-s − 0.585·14-s + (−1.41 + i)15-s + 16-s + 2.82i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.577 − 0.816i)3-s + 0.5·4-s − 0.447i·5-s + (0.408 + 0.577i)6-s + 0.221·7-s − 0.353·8-s + (−0.333 + 0.942i)9-s + 0.316i·10-s + 1.63i·11-s + (−0.288 − 0.408i)12-s + 0.621i·13-s − 0.156·14-s + (−0.365 + 0.258i)15-s + 0.250·16-s + 0.685i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.766397 + 0.148984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.766397 + 0.148984i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 5 | \( 1 + iT \) |
| 19 | \( 1 + (-4.24 - i)T \) |
good | 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 - 5.41iT - 11T^{2} \) |
| 13 | \( 1 - 2.24iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 31 | \( 1 - 6.82iT - 31T^{2} \) |
| 37 | \( 1 - 2.24iT - 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 - 3.17iT - 47T^{2} \) |
| 53 | \( 1 - 3.17T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 4.48T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 2.48T + 73T^{2} \) |
| 79 | \( 1 - 13.3iT - 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 6.58iT - 97T^{2} \) |
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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
−10.79837892818962674749590845992, −9.910551364808147648803213531948, −9.015711564670471186975101553585, −8.002491175347816181301896050313, −7.28563600338662704195527734285, −6.51095389077325742398044562978, −5.37646944716733884808360809484, −4.33966130192215834362048942823, −2.31484195027995799611342602018, −1.30297792593095552895869422482,
0.67745181899837192652731917851, 2.88520835534090871236014342005, 3.81374848090935712154043440781, 5.47284407434638017747440805165, 5.88946852933604129609279160345, 7.21890197136782646176877375883, 8.085965246460928168234430137183, 9.231015380080341790886389765662, 9.698302799989182034805350669125, 10.86535774528953384115027826604