L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.142 − 0.989i)4-s + (−0.415 − 0.909i)5-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (−0.959 − 0.281i)10-s + (0.654 + 0.755i)11-s + (−0.959 − 0.281i)13-s + (−0.415 + 0.909i)14-s + (−0.959 + 0.281i)16-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.841 + 0.540i)20-s + 22-s + (−0.654 + 0.755i)25-s + (−0.841 + 0.540i)26-s + ⋯ |
L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.142 − 0.989i)4-s + (−0.415 − 0.909i)5-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (−0.959 − 0.281i)10-s + (0.654 + 0.755i)11-s + (−0.959 − 0.281i)13-s + (−0.415 + 0.909i)14-s + (−0.959 + 0.281i)16-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.841 + 0.540i)20-s + 22-s + (−0.654 + 0.755i)25-s + (−0.841 + 0.540i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01594480590 - 1.272495347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01594480590 - 1.272495347i\) |
\(L(1)\) |
\(\approx\) |
\(0.7837627152 - 0.7685873589i\) |
\(L(1)\) |
\(\approx\) |
\(0.7837627152 - 0.7685873589i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 11 | \( 1 + (0.654 + 0.755i)T \) |
| 13 | \( 1 + (-0.959 - 0.281i)T \) |
| 17 | \( 1 + (0.142 - 0.989i)T \) |
| 19 | \( 1 + (-0.142 - 0.989i)T \) |
| 29 | \( 1 + (0.142 - 0.989i)T \) |
| 31 | \( 1 + (0.841 + 0.540i)T \) |
| 37 | \( 1 + (0.415 - 0.909i)T \) |
| 41 | \( 1 + (-0.415 - 0.909i)T \) |
| 43 | \( 1 + (0.841 - 0.540i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.959 - 0.281i)T \) |
| 59 | \( 1 + (0.959 + 0.281i)T \) |
| 61 | \( 1 + (0.841 + 0.540i)T \) |
| 67 | \( 1 + (-0.654 + 0.755i)T \) |
| 71 | \( 1 + (0.654 - 0.755i)T \) |
| 73 | \( 1 + (-0.142 - 0.989i)T \) |
| 79 | \( 1 + (-0.959 - 0.281i)T \) |
| 83 | \( 1 + (-0.415 + 0.909i)T \) |
| 89 | \( 1 + (-0.841 + 0.540i)T \) |
| 97 | \( 1 + (0.415 + 0.909i)T \) |
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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
−32.16344838284954587665783056038, −31.26469442806450195980079649119, −30.04339631030474722019608148535, −29.337733195682958425522369467391, −27.31724711616952529911063053136, −26.47445006493770391099542146705, −25.60513181663884056564485449024, −24.34674070358077011849598390880, −23.23900448479697628762224924040, −22.34995446552456728188311829149, −21.575388933814025159687439388786, −19.7439137645674778133185956490, −18.74075061197156891540275766270, −17.118828335972164244771336535112, −16.24239408969097248060333140893, −14.9374796744864671631532386940, −14.13253291384912185478999402853, −12.76391715899163987847729538844, −11.5794178488577670335041284259, −9.95843621077639631021960030837, −8.2291075202367752310088511715, −6.906072922648469822925374419008, −6.060143741768380347643686675204, −4.07462061953857083987613051575, −3.026984064622387764783315125939,
0.51307809705519744920004310773, 2.54014861416951217054938350095, 4.154275150075052354092237916, 5.306980499881969333617715184469, 6.97395205662791469023523599964, 9.08103237246441801027863302932, 9.92529773152285568214862800851, 11.75619872633928672757989594822, 12.44433488236117129511384663823, 13.5028138347521609123744817626, 15.0273612249198926735349714103, 16.045479103773865935839169112753, 17.569081949740993339935814076461, 19.308229163063601392944789402806, 19.827840674393859137580737095907, 20.9550254965658679386908291211, 22.28990000779017034752595968588, 23.01430154772086575279032811494, 24.33419472847554624393332950641, 25.20961794025842199812981198985, 27.04199218396655859066801616622, 28.108565978211339278654113702603, 28.85538862428073564103723275823, 29.95308631248031828432446791918, 31.14676286446958129704826692816