L(s) = 1 | + (−0.809 − 0.587i)3-s + (−0.309 + 0.951i)5-s + (0.309 + 0.951i)9-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)19-s − 23-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (−0.809 + 0.587i)29-s + (0.309 + 0.951i)31-s + (−0.809 + 0.587i)37-s + (−0.309 + 0.951i)39-s + (0.809 + 0.587i)41-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)3-s + (−0.309 + 0.951i)5-s + (0.309 + 0.951i)9-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)19-s − 23-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (−0.809 + 0.587i)29-s + (0.309 + 0.951i)31-s + (−0.809 + 0.587i)37-s + (−0.309 + 0.951i)39-s + (0.809 + 0.587i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07227067841 + 0.2317647475i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07227067841 + 0.2317647475i\) |
\(L(1)\) |
\(\approx\) |
\(0.5799591020 + 0.05104825899i\) |
\(L(1)\) |
\(\approx\) |
\(0.5799591020 + 0.05104825899i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 13 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.809 + 0.587i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.309 - 0.951i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.309 - 0.951i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.5966177061335378757219840459, −24.039335322669045648047679657266, −23.0940461150281976845373073678, −22.312435964274756903702676824988, −21.20395682894652892547920029822, −20.711450590863593378593606758088, −19.596006150897185312020143326654, −18.52160934142274249566801177661, −17.4069568296416638697049265340, −16.647263635467249240404466302600, −16.04020036298826075089965023397, −15.10028503275121233415114128932, −13.876707283331009883936925605871, −12.65383842524029536807441675947, −11.88209934753796588397482336631, −11.14115597048923478443469715139, −9.81533576778262058026497357527, −9.16738696622727076914659633788, −7.93572545589152966570838993849, −6.62895027362622426211887114859, −5.55542804454039835950261073616, −4.557544935015172141852964438132, −3.85366349887086904176647513665, −1.88482683847848500990410569648, −0.16733890167962566418896811946,
1.76611216691672381635299884120, 3.04218788297731916851010772617, 4.440818028261188311624023939311, 5.746960419678808260861674642095, 6.58889237577995712359635431076, 7.486948824105569733571593839979, 8.44473774764182607559810899586, 10.23728680730680028174575358622, 10.723521281408551296473064970552, 11.74653818467986615780629190810, 12.64565303274242593420948465889, 13.562425208787573595987657566858, 14.77822737260943587794484263332, 15.552990786300042783724096082790, 16.72535783451016945844248356703, 17.70621349460551681240104354239, 18.23369466792653659898734180174, 19.31522579694990012611384442579, 19.89931898918920332133316455754, 21.55871705869915560911543173119, 22.167339803544917768816069841442, 22.995336825311992397668948364170, 23.72358840484938383737044589645, 24.58565240546243158290056850755, 25.65612709493920517370190673036