L(s) = 1 | + (0.997 − 0.0654i)3-s + (−0.751 + 0.659i)5-s + (0.991 − 0.130i)9-s + (−0.896 + 0.442i)11-s + (0.195 − 0.980i)13-s + (−0.707 + 0.707i)15-s + (0.965 − 0.258i)17-s + (−0.321 − 0.946i)19-s + (0.130 + 0.991i)23-s + (0.130 − 0.991i)25-s + (0.980 − 0.195i)27-s + (0.831 − 0.555i)29-s + (0.866 + 0.5i)31-s + (−0.866 + 0.5i)33-s + (0.659 + 0.751i)37-s + ⋯ |
L(s) = 1 | + (0.997 − 0.0654i)3-s + (−0.751 + 0.659i)5-s + (0.991 − 0.130i)9-s + (−0.896 + 0.442i)11-s + (0.195 − 0.980i)13-s + (−0.707 + 0.707i)15-s + (0.965 − 0.258i)17-s + (−0.321 − 0.946i)19-s + (0.130 + 0.991i)23-s + (0.130 − 0.991i)25-s + (0.980 − 0.195i)27-s + (0.831 − 0.555i)29-s + (0.866 + 0.5i)31-s + (−0.866 + 0.5i)33-s + (0.659 + 0.751i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.875702486 + 0.2041844752i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.875702486 + 0.2041844752i\) |
\(L(1)\) |
\(\approx\) |
\(1.354815299 + 0.08310619596i\) |
\(L(1)\) |
\(\approx\) |
\(1.354815299 + 0.08310619596i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.997 - 0.0654i)T \) |
| 5 | \( 1 + (-0.751 + 0.659i)T \) |
| 11 | \( 1 + (-0.896 + 0.442i)T \) |
| 13 | \( 1 + (0.195 - 0.980i)T \) |
| 17 | \( 1 + (0.965 - 0.258i)T \) |
| 19 | \( 1 + (-0.321 - 0.946i)T \) |
| 23 | \( 1 + (0.130 + 0.991i)T \) |
| 29 | \( 1 + (0.831 - 0.555i)T \) |
| 31 | \( 1 + (0.866 + 0.5i)T \) |
| 37 | \( 1 + (0.659 + 0.751i)T \) |
| 41 | \( 1 + (0.923 + 0.382i)T \) |
| 43 | \( 1 + (0.555 - 0.831i)T \) |
| 47 | \( 1 + (-0.258 + 0.965i)T \) |
| 53 | \( 1 + (-0.896 + 0.442i)T \) |
| 59 | \( 1 + (0.946 + 0.321i)T \) |
| 61 | \( 1 + (-0.442 + 0.896i)T \) |
| 67 | \( 1 + (0.997 - 0.0654i)T \) |
| 71 | \( 1 + (0.382 + 0.923i)T \) |
| 73 | \( 1 + (-0.608 - 0.793i)T \) |
| 79 | \( 1 + (-0.965 - 0.258i)T \) |
| 83 | \( 1 + (-0.980 - 0.195i)T \) |
| 89 | \( 1 + (0.793 + 0.608i)T \) |
| 97 | \( 1 - iT \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.486198668501678891492559990313, −21.06932374839831226307118095461, −20.44958411605010796725852548835, −19.464791479529696559569556776989, −18.95430356093231652572038278952, −18.32028125938774241140954682136, −16.86674396178046777249744261580, −16.18814197414245857934601754032, −15.68076986149597084766662580204, −14.55081636657448790709580479694, −14.1166585258832685875447920955, −12.91242918016191564352241335238, −12.5237048516885841473960728221, −11.42124894927530737005697598994, −10.40617386076939639641136751442, −9.55569691620932201602641905341, −8.5220216932911408783429860912, −8.15281964146731230815849197700, −7.31057902176646412414410283420, −6.11523884269675577631300281456, −4.862612341083446504816874311881, −4.10029994309600720839977920109, −3.26050742012827401110406066150, −2.189686553228058740511707642060, −0.98746833326672135657307701482,
1.02162636904296961313368179191, 2.68130224311246083411545665386, 2.94254621247923989624640894572, 4.07909206620764751223148899169, 5.024034909154488324256282507035, 6.34110094837483973287259236841, 7.4904494783373894031687405107, 7.76702480520236574429746779494, 8.66877799435894904409838168827, 9.855899666121149094748291402333, 10.40078710307363145596225674584, 11.42243092917656902761014505250, 12.44268178187612880024696768327, 13.178707306970862082791350690836, 14.03300998639617974385137864935, 14.86682601065406699427490730358, 15.60591534182497888272741055515, 15.87636872067537874347549998695, 17.47794022121288727950667480651, 18.1469194457872180513351142623, 19.01505202032046226628914454513, 19.52925585938418792445025997507, 20.35136281800380831089791132353, 21.05399730906875860093734948441, 21.86368697977507241853057341830