L(s) = 1 | + (0.989 + 0.142i)3-s + (−0.654 − 0.755i)7-s + (0.959 + 0.281i)9-s + (0.540 − 0.841i)11-s + (0.755 + 0.654i)13-s + (−0.415 − 0.909i)17-s + (0.909 + 0.415i)19-s + (−0.540 − 0.841i)21-s + (0.909 + 0.415i)27-s + (−0.909 + 0.415i)29-s + (−0.142 − 0.989i)31-s + (0.654 − 0.755i)33-s + (0.281 − 0.959i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯ |
L(s) = 1 | + (0.989 + 0.142i)3-s + (−0.654 − 0.755i)7-s + (0.959 + 0.281i)9-s + (0.540 − 0.841i)11-s + (0.755 + 0.654i)13-s + (−0.415 − 0.909i)17-s + (0.909 + 0.415i)19-s + (−0.540 − 0.841i)21-s + (0.909 + 0.415i)27-s + (−0.909 + 0.415i)29-s + (−0.142 − 0.989i)31-s + (0.654 − 0.755i)33-s + (0.281 − 0.959i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.191456438 - 0.8915606856i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.191456438 - 0.8915606856i\) |
\(L(1)\) |
\(\approx\) |
\(1.499831494 - 0.2040770763i\) |
\(L(1)\) |
\(\approx\) |
\(1.499831494 - 0.2040770763i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.989 + 0.142i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 11 | \( 1 + (0.540 - 0.841i)T \) |
| 13 | \( 1 + (0.755 + 0.654i)T \) |
| 17 | \( 1 + (-0.415 - 0.909i)T \) |
| 19 | \( 1 + (0.909 + 0.415i)T \) |
| 29 | \( 1 + (-0.909 + 0.415i)T \) |
| 31 | \( 1 + (-0.142 - 0.989i)T \) |
| 37 | \( 1 + (0.281 - 0.959i)T \) |
| 41 | \( 1 + (0.959 - 0.281i)T \) |
| 43 | \( 1 + (-0.989 - 0.142i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.755 - 0.654i)T \) |
| 59 | \( 1 + (0.755 + 0.654i)T \) |
| 61 | \( 1 + (0.989 - 0.142i)T \) |
| 67 | \( 1 + (-0.540 - 0.841i)T \) |
| 71 | \( 1 + (-0.841 + 0.540i)T \) |
| 73 | \( 1 + (0.415 - 0.909i)T \) |
| 79 | \( 1 + (-0.654 + 0.755i)T \) |
| 83 | \( 1 + (-0.281 + 0.959i)T \) |
| 89 | \( 1 + (0.142 - 0.989i)T \) |
| 97 | \( 1 + (0.959 - 0.281i)T \) |
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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
−20.09342304669125434781186246252, −19.633705536618369103421682616, −18.84926269578888582870591036561, −18.12282092627697920781649236733, −17.54156667498536874054921470879, −16.322570507370372584072327281695, −15.666093959138988226776630822971, −15.03137388996766005768122214603, −14.51848985736373763516560718293, −13.31081806990555715440227310368, −13.06201165252650884963454377073, −12.213431095971146208124744303646, −11.37331128577232676939703016382, −10.185573968791289078661244289767, −9.64358384025659484254282063551, −8.84706837606673133375552881841, −8.292727719787530416684574498233, −7.30770332844725274107854911729, −6.57891333882432068785070886242, −5.74384454977551291178564322, −4.62681373977270163981984740825, −3.65646865286049763705484555041, −3.04250346471555569575082571588, −2.08817201716465058338106498252, −1.23310633871469743554259181001,
0.7922384909861583856714940133, 1.82340982564789899268362284624, 2.93787262628767672510717588503, 3.70627543508551745034423215932, 4.14437285152805266971752418617, 5.41629569158424438721750777307, 6.45722895266988372580775341524, 7.16864980713924630903365492256, 7.89328219917016841777836783194, 8.925211746755261266159878068661, 9.345643263511796624270055215073, 10.11886906922202676176213375308, 11.091091102124633271582223907196, 11.72570207768985482491796103057, 13.03918775054298492775429010132, 13.410204378645101699696139283463, 14.13492416455107681499747551396, 14.654782607735260431898420952632, 15.858915776703835915558635255622, 16.23717462416444342596529431090, 16.836355995371194289258681623461, 18.15103457892657928060793689289, 18.650886424752322481721765939710, 19.5017197255836417090791767451, 19.96207711767773449375443705361