L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.766 − 0.642i)10-s + (0.5 + 0.866i)11-s + (0.939 − 0.342i)13-s + (0.173 + 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s − 20-s + (0.939 + 0.342i)22-s + (−0.173 + 0.984i)23-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.766 − 0.642i)10-s + (0.5 + 0.866i)11-s + (0.939 − 0.342i)13-s + (0.173 + 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s − 20-s + (0.939 + 0.342i)22-s + (−0.173 + 0.984i)23-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9767511401 - 0.7240934479i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9767511401 - 0.7240934479i\) |
\(L(1)\) |
\(\approx\) |
\(1.202616685 - 0.5966948384i\) |
\(L(1)\) |
\(\approx\) |
\(1.202616685 - 0.5966948384i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.939 - 0.342i)T \) |
| 17 | \( 1 + (-0.766 + 0.642i)T \) |
| 23 | \( 1 + (-0.173 + 0.984i)T \) |
| 29 | \( 1 + (0.766 + 0.642i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 + (0.173 + 0.984i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 + (0.173 - 0.984i)T \) |
| 59 | \( 1 + (0.766 - 0.642i)T \) |
| 61 | \( 1 + (0.173 - 0.984i)T \) |
| 67 | \( 1 + (-0.766 - 0.642i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + (-0.939 - 0.342i)T \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 97 | \( 1 + (-0.766 + 0.642i)T \) |
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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
−33.12251750024732692882093875921, −32.23811050457515754264183524501, −30.85952212499981240506074857458, −30.12711441603230070099260780331, −29.08330768633168966646313716740, −26.94518064586343169915945639911, −26.39901131709837712824127124542, −25.23082423092246179398228967895, −23.90117265168570621862572843080, −22.89763745434151662563110171146, −22.14434593921332734753186482834, −20.80121938811289109863826944851, −19.34722399455726448170423429061, −17.92550290524197437959346215209, −16.54257296197703713137949413541, −15.60520423219151143759517762256, −14.115053591798576692528383729876, −13.54992984075997721751113328659, −11.7725497887800773966645219789, −10.6108711247087023108230919278, −8.57656934094403387390761687380, −6.986347355265703604031475910026, −6.2588285258460393877005356274, −4.18971569348015419628187312811, −3.05264398190378724706401307006,
1.7658686899899328616425696788, 3.67571423190668730345682445197, 5.07560399290877310363568765270, 6.37963916873638088968906029163, 8.6648661256333105979411848080, 9.83945010535109530721733788189, 11.54452145669419879389219900533, 12.53618834464391696137203710792, 13.40881044129993141359404765825, 15.13195040821131067147850075480, 15.953687904634736807773296308304, 17.74950014902260358206666857757, 19.27196170744109415812590917122, 20.1630978287607629183099080210, 21.24586818531810739534993203841, 22.37654099363572910779026619177, 23.46222524327603173494161377350, 24.61700250965743683097795208460, 25.583302238983055944401736249774, 27.78304659129735242585904889748, 28.2217007642433886842942631201, 29.30042259031889790386309744726, 30.72166075342355603906820324253, 31.48275137764504562511800644629, 32.563371582925584937973189138047