L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s − i·8-s + (0.5 − 0.866i)11-s + i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s − i·22-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)26-s + 29-s + (−0.5 + 0.866i)31-s + (−0.866 − 0.5i)32-s − 34-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s − i·8-s + (0.5 − 0.866i)11-s + i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s − i·22-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)26-s + 29-s + (−0.5 + 0.866i)31-s + (−0.866 − 0.5i)32-s − 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.410458732 - 0.7832199219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.410458732 - 0.7832199219i\) |
\(L(1)\) |
\(\approx\) |
\(1.476677464 - 0.5508734754i\) |
\(L(1)\) |
\(\approx\) |
\(1.476677464 - 0.5508734754i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + iT \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (-0.866 - 0.5i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 - iT \) |
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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
−30.32650777316238144146570611270, −29.02943013452896599173220140468, −27.838190870733763052872113655299, −26.51954950127719280450784265811, −25.57241171385210503886618888650, −24.64921977641801101620989366933, −23.71263176298932318278132996458, −22.5295454222600922196692694296, −21.97792366401787458543062961326, −20.518222745565674022213097871012, −19.83617513541340213850209476408, −17.93413964529868432316625586786, −17.17875149364298899807977253053, −15.76751997774175786579290694964, −15.0563038951139395597528862812, −13.85625375405700939121798128639, −12.79352813170290257454875265306, −11.82702345896763232158386241644, −10.44825955166816130530406605621, −8.77536192791692769784513965787, −7.47900107379176058858088424133, −6.38405566604915437999773174333, −5.055763409223087369159679222858, −3.8658184923186593132098105955, −2.3195937116227977530828456743,
1.638806404501833589960220336230, 3.25328499630752197132259031896, 4.45809640370228121992385795147, 5.856895453491883306755942201746, 6.96354068215842024260311020616, 8.82623242480204873480352443138, 10.132214293627130218730733273, 11.41040042735419276322193869723, 12.14635731430072303659474224317, 13.679846001335989603256430814981, 14.18802553599217821394137154308, 15.64148429718215530826827688273, 16.5671194199174155489468749619, 18.25913594425191395036397999649, 19.32348136313820219517260261900, 20.234195207005229076373620793985, 21.44623703596392182130736102607, 22.11314081836906335726843881975, 23.29240258211898176313458610974, 24.23792121437381967965878378902, 25.08306896335298695034282979970, 26.59445275256492813903231123616, 27.65614632577086591711433770383, 28.937446343019251696345930602042, 29.45986243088741434452924010482