L(s) = 1 | + (0.906 − 0.422i)5-s + (−0.422 + 0.906i)11-s + (0.573 − 0.819i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.984 − 0.173i)23-s + (0.642 − 0.766i)25-s + (−0.819 + 0.573i)29-s + (0.766 − 0.642i)31-s + (0.258 + 0.965i)37-s + (0.984 − 0.173i)41-s + (0.996 − 0.0871i)43-s + (−0.766 − 0.642i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯ |
L(s) = 1 | + (0.906 − 0.422i)5-s + (−0.422 + 0.906i)11-s + (0.573 − 0.819i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.984 − 0.173i)23-s + (0.642 − 0.766i)25-s + (−0.819 + 0.573i)29-s + (0.766 − 0.642i)31-s + (0.258 + 0.965i)37-s + (0.984 − 0.173i)41-s + (0.996 − 0.0871i)43-s + (−0.766 − 0.642i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.642201911 - 0.1194948607i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.642201911 - 0.1194948607i\) |
\(L(1)\) |
\(\approx\) |
\(1.416422297 - 0.05533827221i\) |
\(L(1)\) |
\(\approx\) |
\(1.416422297 - 0.05533827221i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.906 - 0.422i)T \) |
| 11 | \( 1 + (-0.422 + 0.906i)T \) |
| 13 | \( 1 + (0.573 - 0.819i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
| 23 | \( 1 + (0.984 - 0.173i)T \) |
| 29 | \( 1 + (-0.819 + 0.573i)T \) |
| 31 | \( 1 + (0.766 - 0.642i)T \) |
| 37 | \( 1 + (0.258 + 0.965i)T \) |
| 41 | \( 1 + (0.984 - 0.173i)T \) |
| 43 | \( 1 + (0.996 - 0.0871i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 + (-0.258 - 0.965i)T \) |
| 59 | \( 1 + (0.819 + 0.573i)T \) |
| 61 | \( 1 + (0.0871 + 0.996i)T \) |
| 67 | \( 1 + (-0.422 - 0.906i)T \) |
| 71 | \( 1 + (0.866 - 0.5i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.819 - 0.573i)T \) |
| 89 | \( 1 - iT \) |
| 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
−17.64584171419156703221021482967, −17.1885164781771637857093268540, −16.26302257191166293154247594870, −15.98861789688037055057367896203, −14.99199880264091853855817927310, −14.24164301931761185799336112718, −13.89722145050357011677490526277, −13.19960694560998621496494585611, −12.634353920042236223650112204213, −11.54675098284364094350061568137, −11.10165397534215657547588999937, −10.533379361791986716749582030717, −9.536204460445370791769990723, −9.29065128865618884533528176902, −8.416997665323758208448300024746, −7.56448724092686259792879209993, −6.93200544272686058452269088942, −6.08147727539971781107926967747, −5.65632389827337423427372863570, −4.9133936928310044606193671308, −3.92253552790078105364792775608, −3.05159017726523336827464853215, −2.615269136648296655498718006914, −1.52998943241300154544019036157, −0.8580408899202381644197403800,
0.89944153854752048990129517685, 1.4605485750034990057204897438, 2.43432877485902020223067396005, 3.10621106213593193223380804258, 4.00343070799628574541114995558, 4.97509769079163839971130919069, 5.44388593967538905464586124928, 6.03741939661195132089403123043, 6.93073529865840548811041906018, 7.723228458649997205967561988038, 8.26727690046294463306368628329, 9.181274373932172317246936240598, 9.77147167932168216102593379919, 10.26895728712457180834838603355, 10.9435964226094533007028431431, 11.93131510062322155193775781071, 12.51778092658090354675660855864, 13.175944327338963718450934376411, 13.54496257649166744466729064697, 14.62854053432242183736405617530, 14.838871945709176737092928505581, 15.8737100132897620224270487461, 16.3881667846787904210930980974, 17.13084208620321230947627363642, 17.640353564748686923683494420104