L(s) = 1 | + (0.612 + 0.790i)3-s + (−0.809 − 0.587i)7-s + (−0.248 + 0.968i)9-s + (0.995 − 0.0941i)11-s + (0.509 + 0.860i)13-s + (−0.982 − 0.187i)17-s + (0.612 − 0.790i)19-s + (−0.0314 − 0.999i)21-s + (0.535 + 0.844i)23-s + (−0.917 + 0.397i)27-s + (0.338 + 0.940i)29-s + (−0.187 + 0.982i)31-s + (0.684 + 0.728i)33-s + (−0.397 + 0.917i)37-s + (−0.368 + 0.929i)39-s + ⋯ |
L(s) = 1 | + (0.612 + 0.790i)3-s + (−0.809 − 0.587i)7-s + (−0.248 + 0.968i)9-s + (0.995 − 0.0941i)11-s + (0.509 + 0.860i)13-s + (−0.982 − 0.187i)17-s + (0.612 − 0.790i)19-s + (−0.0314 − 0.999i)21-s + (0.535 + 0.844i)23-s + (−0.917 + 0.397i)27-s + (0.338 + 0.940i)29-s + (−0.187 + 0.982i)31-s + (0.684 + 0.728i)33-s + (−0.397 + 0.917i)37-s + (−0.368 + 0.929i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4932398316 + 2.061623582i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4932398316 + 2.061623582i\) |
\(L(1)\) |
\(\approx\) |
\(1.138054658 + 0.4643726960i\) |
\(L(1)\) |
\(\approx\) |
\(1.138054658 + 0.4643726960i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.612 + 0.790i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (0.995 - 0.0941i)T \) |
| 13 | \( 1 + (0.509 + 0.860i)T \) |
| 17 | \( 1 + (-0.982 - 0.187i)T \) |
| 19 | \( 1 + (0.612 - 0.790i)T \) |
| 23 | \( 1 + (0.535 + 0.844i)T \) |
| 29 | \( 1 + (0.338 + 0.940i)T \) |
| 31 | \( 1 + (-0.187 + 0.982i)T \) |
| 37 | \( 1 + (-0.397 + 0.917i)T \) |
| 41 | \( 1 + (-0.844 - 0.535i)T \) |
| 43 | \( 1 + (0.891 + 0.453i)T \) |
| 47 | \( 1 + (0.481 + 0.876i)T \) |
| 53 | \( 1 + (0.999 - 0.0314i)T \) |
| 59 | \( 1 + (0.661 - 0.750i)T \) |
| 61 | \( 1 + (-0.975 - 0.218i)T \) |
| 67 | \( 1 + (0.338 - 0.940i)T \) |
| 71 | \( 1 + (0.481 + 0.876i)T \) |
| 73 | \( 1 + (0.0627 - 0.998i)T \) |
| 79 | \( 1 + (-0.992 - 0.125i)T \) |
| 83 | \( 1 + (-0.790 - 0.612i)T \) |
| 89 | \( 1 + (-0.998 - 0.0627i)T \) |
| 97 | \( 1 + (0.904 + 0.425i)T \) |
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
−18.36632344061723509543999703849, −17.38376743380097664280310207982, −16.79920307464630904031133727496, −15.808907283437171677634302206853, −15.244274089970574494684438908867, −14.65049684511876088331276277586, −13.80059257048029961757423605525, −13.2321570381002633420492922505, −12.61062803375135323368581380757, −12.01636911414456218902616960791, −11.34304795814231715316972652165, −10.270758421906645480113059868419, −9.58149760765179397857367028730, −8.763450547418345762700825457679, −8.48765983355823323484491272858, −7.43171027411390374530163252955, −6.8004370322423058968248048699, −6.07865014807972805202124280952, −5.59994766167554741247509605101, −4.15223025190392196783347969832, −3.61663047435806084310015543636, −2.7102238598544434548899494471, −2.12422089079182557337687252004, −1.072520535297860873327196459037, −0.31425580310676362286170459873,
1.00350394598944353458369988381, 1.88250196609733818324971555875, 3.0714812408887650425125087368, 3.44239572503473789643693837450, 4.309222597830320378590829547712, 4.84991037510983294689805158683, 5.92110469047885761840307482071, 6.939140564474534505143054215709, 7.10909695138633743317981422641, 8.44696643416095630921939556470, 9.085984926398051721356953984619, 9.3781624883950988234235457375, 10.24174130105432306199524222759, 11.027551920688906665698733639323, 11.489705946503919691474282326826, 12.519922252013941398293350115633, 13.43815585521405380509216335705, 13.86541131713434599280687570891, 14.359659591890702897843330332582, 15.44252303902389553454587734077, 15.77777183589332778083653230592, 16.49038706453275485520421647533, 17.07061712969906442789855960328, 17.802379895054151874786370510188, 18.863390928268321111225497294