L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.733 − 0.680i)5-s + (−0.222 − 0.974i)8-s + (−0.955 + 0.294i)10-s + (0.826 − 0.563i)11-s + (−0.826 + 0.563i)13-s + (−0.222 + 0.974i)16-s + (0.988 + 0.149i)17-s + (0.5 − 0.866i)19-s + (0.988 + 0.149i)20-s + (−0.988 + 0.149i)22-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.988 − 0.149i)26-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.733 − 0.680i)5-s + (−0.222 − 0.974i)8-s + (−0.955 + 0.294i)10-s + (0.826 − 0.563i)11-s + (−0.826 + 0.563i)13-s + (−0.222 + 0.974i)16-s + (0.988 + 0.149i)17-s + (0.5 − 0.866i)19-s + (0.988 + 0.149i)20-s + (−0.988 + 0.149i)22-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.988 − 0.149i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7439247998 - 1.174640996i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7439247998 - 1.174640996i\) |
\(L(1)\) |
\(\approx\) |
\(0.7912940546 - 0.3528785410i\) |
\(L(1)\) |
\(\approx\) |
\(0.7912940546 - 0.3528785410i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.900 - 0.433i)T \) |
| 5 | \( 1 + (0.733 - 0.680i)T \) |
| 11 | \( 1 + (0.826 - 0.563i)T \) |
| 13 | \( 1 + (-0.826 + 0.563i)T \) |
| 17 | \( 1 + (0.988 + 0.149i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.365 - 0.930i)T \) |
| 29 | \( 1 + (-0.988 - 0.149i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.365 + 0.930i)T \) |
| 41 | \( 1 + (-0.955 - 0.294i)T \) |
| 43 | \( 1 + (0.955 - 0.294i)T \) |
| 47 | \( 1 + (0.900 + 0.433i)T \) |
| 53 | \( 1 + (0.365 - 0.930i)T \) |
| 59 | \( 1 + (0.222 - 0.974i)T \) |
| 61 | \( 1 + (-0.623 + 0.781i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.623 + 0.781i)T \) |
| 73 | \( 1 + (-0.826 - 0.563i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.826 - 0.563i)T \) |
| 89 | \( 1 + (-0.0747 + 0.997i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−24.482203629539939111545433305657, −23.257799943208163420505663310981, −22.5316155453839183505178611579, −21.53160565211670919106543272022, −20.47378859694276143359158387942, −19.697729326401443242518118209373, −18.7201460427235784146355783458, −18.084844085609496274422543722543, −17.16312975500595430819561353365, −16.683430084548540444556065502235, −15.33344006088205721711719874279, −14.63600882787879808399100998093, −14.02384690833502358692906564002, −12.56073872565645486267921611984, −11.52698857141112722710177389831, −10.49423005142310583526109078497, −9.71679270384396846227666751923, −9.1677273987090204070201998746, −7.58557324936445829997529796412, −7.22453172101273887625942902319, −5.94062057407851337127858092790, −5.31653601248257260731580650128, −3.47651079058731693714748692824, −2.21156214175028145779747190775, −1.21537497455728830964926026936,
0.53590095557196448651264508645, 1.55310429376998921264000296497, 2.64141904528819469191085751171, 3.91991041515304563976546651895, 5.239155575734119306844428910117, 6.430967195153587215262688147738, 7.39812618901562972667955318083, 8.598823613698685237728545658088, 9.278583780642095312605849814507, 9.95275979253405692422891685597, 11.09684267728123460896080378726, 12.00579704251742733209865050612, 12.76469778089984600103551744207, 13.825763010335866391424189634788, 14.8469926987540406559267714493, 16.23564751997951123302574246489, 16.84806304242071288239325236725, 17.35338989120180013417887749692, 18.502955030408911923355791541357, 19.20312705624305312795456113950, 20.168359534809999660692246712765, 20.80077391057541702326043723557, 21.8110137375300586337233981748, 22.21693324416311750904677593754, 23.99877597419965191309237809861