L(s) = 1 | − i·7-s − i·11-s + (−0.5 + 0.866i)13-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)23-s + (0.866 + 0.5i)29-s + 31-s + 37-s + (0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.866 + 0.5i)47-s − 49-s + (−0.5 + 0.866i)53-s + (−0.866 + 0.5i)59-s + (−0.866 − 0.5i)61-s + ⋯ |
L(s) = 1 | − i·7-s − i·11-s + (−0.5 + 0.866i)13-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)23-s + (0.866 + 0.5i)29-s + 31-s + 37-s + (0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.866 + 0.5i)47-s − 49-s + (−0.5 + 0.866i)53-s + (−0.866 + 0.5i)59-s + (−0.866 − 0.5i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.862220601 - 0.05866348185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.862220601 - 0.05866348185i\) |
\(L(1)\) |
\(\approx\) |
\(1.141885453 - 0.09297534009i\) |
\(L(1)\) |
\(\approx\) |
\(1.141885453 - 0.09297534009i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 - iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.866 + 0.5i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.866 + 0.5i)T \) |
| 61 | \( 1 + (-0.866 - 0.5i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−18.26720843561574592808950019017, −17.421236275763044474376827257130, −17.171888261172293518067717951203, −16.08843951787043342772120792856, −15.4307472709292657268210501148, −14.96029138525061618592239347777, −14.40843782972686563705198492410, −13.44143645740433692362744176903, −12.547328366083423241517448080387, −12.34518106201916128711356662278, −11.63113924576402434376827106287, −10.57215150238251939971955211468, −10.102878800084575114751606731152, −9.34910147672630754342625233844, −8.635026294070718284865205243741, −7.88259368270459610027048322994, −7.284668572960575163883026782, −6.31248083702656438385702679778, −5.697149452366561698715995119326, −4.93307995274664551342037118665, −4.30676635773954017458480780605, −3.13129387543713544034733401779, −2.59627968246964616520340606053, −1.78044285249449875465781644295, −0.66952996976114763802309166535,
0.87292066293965371988073685925, 1.3461441056579436903152479763, 2.810811764082341965810314610395, 3.15048155072699127790188904022, 4.33368617750231884203996682709, 4.66594484915310404945807552293, 5.77688181669642168759496709434, 6.406678002954345340814014896946, 7.28291243459093359804648582567, 7.72904554919162769886027241597, 8.61812401560635679550093687261, 9.42875135702495466596082721230, 9.96478563013947593593384227525, 10.87517830703348254823214323429, 11.307477596859582026295028138787, 12.10166504653562208581387722916, 12.85834283036578673026559962159, 13.75660976075128858795079195589, 14.0161153735780445255292415349, 14.70476647246975737276086098178, 15.65274815347996624615298049465, 16.43672008036755361164654466150, 16.735955096819778359079989440530, 17.43596033259564259890881445739, 18.244327194277576292283246089916