L(s) = 1 | + (−0.997 − 0.0697i)2-s + (0.848 + 0.529i)3-s + (0.990 + 0.139i)4-s + (0.559 + 0.829i)5-s + (−0.809 − 0.587i)6-s + (−0.615 + 0.788i)7-s + (−0.978 − 0.207i)8-s + (0.438 + 0.898i)9-s + (−0.5 − 0.866i)10-s + (0.766 + 0.642i)12-s + (−0.719 − 0.694i)13-s + (0.669 − 0.743i)14-s + (0.0348 + 0.999i)15-s + (0.961 + 0.275i)16-s + (−0.719 + 0.694i)17-s + (−0.374 − 0.927i)18-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0697i)2-s + (0.848 + 0.529i)3-s + (0.990 + 0.139i)4-s + (0.559 + 0.829i)5-s + (−0.809 − 0.587i)6-s + (−0.615 + 0.788i)7-s + (−0.978 − 0.207i)8-s + (0.438 + 0.898i)9-s + (−0.5 − 0.866i)10-s + (0.766 + 0.642i)12-s + (−0.719 − 0.694i)13-s + (0.669 − 0.743i)14-s + (0.0348 + 0.999i)15-s + (0.961 + 0.275i)16-s + (−0.719 + 0.694i)17-s + (−0.374 − 0.927i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4832740280 + 0.8843075116i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4832740280 + 0.8843075116i\) |
\(L(1)\) |
\(\approx\) |
\(0.7801477446 + 0.4318742845i\) |
\(L(1)\) |
\(\approx\) |
\(0.7801477446 + 0.4318742845i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.997 - 0.0697i)T \) |
| 3 | \( 1 + (0.848 + 0.529i)T \) |
| 5 | \( 1 + (0.559 + 0.829i)T \) |
| 7 | \( 1 + (-0.615 + 0.788i)T \) |
| 13 | \( 1 + (-0.719 - 0.694i)T \) |
| 17 | \( 1 + (-0.719 + 0.694i)T \) |
| 19 | \( 1 + (0.848 + 0.529i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.669 + 0.743i)T \) |
| 31 | \( 1 + (-0.809 + 0.587i)T \) |
| 41 | \( 1 + (-0.882 + 0.469i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.669 - 0.743i)T \) |
| 53 | \( 1 + (0.559 - 0.829i)T \) |
| 59 | \( 1 + (0.0348 + 0.999i)T \) |
| 61 | \( 1 + (0.438 - 0.898i)T \) |
| 67 | \( 1 + (-0.939 - 0.342i)T \) |
| 71 | \( 1 + (-0.997 + 0.0697i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (0.961 - 0.275i)T \) |
| 83 | \( 1 + (-0.719 + 0.694i)T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 97 | \( 1 + (0.913 + 0.406i)T \) |
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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
−24.18674535151375255112251161911, −23.77289564567706298598946440571, −22.10482784253994797639958667334, −20.993412912231981295197909960524, −20.21476935905606179460570749338, −19.76828159432683680784380050169, −18.92878039866464097696935870026, −17.832512445385384406678531060136, −17.236751306360936711582241893185, −16.22657741627421248953756387912, −15.5026261766091164468032935651, −14.09995000538176697807146363830, −13.51388792546499725021649418912, −12.4479619243844928346763857158, −11.523880614536404932800251315403, −10.00367067213753886897601376282, −9.45427855064692611978039845511, −8.79824743928279493388092773643, −7.54720492318062191946533240656, −7.01901348477506041672646377405, −5.87378911509642522098050441852, −4.27755073275738236292139095378, −2.83920252231617869222416879759, −1.86250133428509910583217548453, −0.69648428273769796121669628309,
1.93121074592152929068934242277, 2.71509863253402432245024904652, 3.50286127254151402020839906389, 5.38509065711966406423531374130, 6.48019169462181266395827386356, 7.4613603705142492979867549016, 8.52352974141101042508489594286, 9.29762662285599188795325514154, 10.18658274553111417897697272999, 10.609464020152885271028920404238, 12.03956984589147694120709248735, 13.04656549193261748154156048326, 14.372512145736234523538517892594, 15.04997550881058524891985843714, 15.812558061797896060294588256114, 16.70070606900859152982885456070, 17.944457081267318807596470373287, 18.47866811746746164227048944708, 19.47681906556098731778216556675, 20.00745327868171871579092083851, 21.07402897414609712280567178305, 21.96984819660282474728415304814, 22.38012223062060166982386149647, 24.20778405935929170940406561080, 25.18929921488622416010976332287