L(s) = 1 | + (−0.0475 − 0.998i)2-s + (0.5 + 0.866i)3-s + (−0.995 + 0.0950i)4-s + (0.786 + 0.618i)5-s + (0.841 − 0.540i)6-s + (0.142 + 0.989i)8-s + (−0.5 + 0.866i)9-s + (0.580 − 0.814i)10-s + (−0.580 − 0.814i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)15-s + (0.981 − 0.189i)16-s + (0.235 + 0.971i)17-s + (0.888 + 0.458i)18-s + (0.235 − 0.971i)19-s + (−0.841 − 0.540i)20-s + ⋯ |
L(s) = 1 | + (−0.0475 − 0.998i)2-s + (0.5 + 0.866i)3-s + (−0.995 + 0.0950i)4-s + (0.786 + 0.618i)5-s + (0.841 − 0.540i)6-s + (0.142 + 0.989i)8-s + (−0.5 + 0.866i)9-s + (0.580 − 0.814i)10-s + (−0.580 − 0.814i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)15-s + (0.981 − 0.189i)16-s + (0.235 + 0.971i)17-s + (0.888 + 0.458i)18-s + (0.235 − 0.971i)19-s + (−0.841 − 0.540i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.758039945 + 0.3320976172i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.758039945 + 0.3320976172i\) |
\(L(1)\) |
\(\approx\) |
\(1.296320185 + 0.02338297586i\) |
\(L(1)\) |
\(\approx\) |
\(1.296320185 + 0.02338297586i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.0475 - 0.998i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.786 + 0.618i)T \) |
| 13 | \( 1 + (0.415 - 0.909i)T \) |
| 17 | \( 1 + (0.235 + 0.971i)T \) |
| 19 | \( 1 + (0.235 - 0.971i)T \) |
| 23 | \( 1 + (0.981 - 0.189i)T \) |
| 29 | \( 1 + (0.959 - 0.281i)T \) |
| 31 | \( 1 + (-0.580 + 0.814i)T \) |
| 37 | \( 1 + (-0.995 - 0.0950i)T \) |
| 41 | \( 1 + (0.841 - 0.540i)T \) |
| 43 | \( 1 + (0.142 + 0.989i)T \) |
| 47 | \( 1 + (0.888 - 0.458i)T \) |
| 53 | \( 1 + (0.981 + 0.189i)T \) |
| 59 | \( 1 + (-0.0475 + 0.998i)T \) |
| 61 | \( 1 + (-0.888 + 0.458i)T \) |
| 67 | \( 1 + (-0.888 - 0.458i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (-0.327 + 0.945i)T \) |
| 79 | \( 1 + (0.786 + 0.618i)T \) |
| 83 | \( 1 + (-0.654 + 0.755i)T \) |
| 89 | \( 1 + (-0.723 + 0.690i)T \) |
| 97 | \( 1 + (0.142 + 0.989i)T \) |
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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
−22.25410841995670896948412638718, −21.12455544293272648403792051035, −20.60788333671688604100201206006, −19.409292532496960600910399168975, −18.63507715252984497677661169908, −18.08943697491722846516781858660, −17.20522525410146868441449337650, −16.56822277789378722740344135218, −15.72213149645180047271339374703, −14.54833913074589340295063266350, −13.9929351084282634026227555437, −13.432754504659623742087995926698, −12.60848418714087366898226736498, −11.79643786850679246543210020586, −10.245928882762590078227753452997, −9.12425817696113572229432261299, −8.95798325997018671473947675927, −7.82456122258031816408903408905, −7.06091029198133862147521323818, −6.20539601925191046748013237144, −5.473286396352731027873000049191, −4.42033294496193433175531898884, −3.20533258866492748222550727538, −1.83356302831284148122717357435, −0.89282022292330986033722579378,
1.29560073068822158570243417516, 2.57498645468339591357650639655, 3.064343109694166327721984161158, 4.03868633334529899482656797174, 5.11440731703649546030962009364, 5.84870501515419460132431578864, 7.328923401163905154216066647435, 8.55640346742754744030414280464, 9.06616167269531125657310990937, 10.08920325652005082634581654725, 10.5950079721010511528129044875, 11.1272454109505191088471776630, 12.456535278679692284813052385660, 13.34795246820209604191940374907, 13.94570423296098320011279102145, 14.809091987152247461507379016745, 15.46757425866872457638255438387, 16.7538249159858653348604290602, 17.56293674761406591120058944250, 18.175130820544181436431447548372, 19.31276914022629609587411951398, 19.73149494716034318007640901592, 20.80915453332091278756022473575, 21.25511345579231279904151093640, 21.91750739087141861411861697651