L(s) = 1 | + (−0.422 − 0.906i)5-s + (−0.906 − 0.422i)11-s + (0.906 − 0.422i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (0.984 − 0.173i)23-s + (−0.642 + 0.766i)25-s + (0.422 − 0.906i)29-s + (0.939 + 0.342i)31-s + (−0.707 − 0.707i)37-s + (0.342 − 0.939i)41-s + (−0.819 + 0.573i)43-s + (0.939 − 0.342i)47-s + (−0.258 + 0.965i)53-s + i·55-s + ⋯ |
L(s) = 1 | + (−0.422 − 0.906i)5-s + (−0.906 − 0.422i)11-s + (0.906 − 0.422i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (0.984 − 0.173i)23-s + (−0.642 + 0.766i)25-s + (0.422 − 0.906i)29-s + (0.939 + 0.342i)31-s + (−0.707 − 0.707i)37-s + (0.342 − 0.939i)41-s + (−0.819 + 0.573i)43-s + (0.939 − 0.342i)47-s + (−0.258 + 0.965i)53-s + i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.04207714095 - 0.5318046579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.04207714095 - 0.5318046579i\) |
\(L(1)\) |
\(\approx\) |
\(0.8013704468 - 0.2369344400i\) |
\(L(1)\) |
\(\approx\) |
\(0.8013704468 - 0.2369344400i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.422 - 0.906i)T \) |
| 11 | \( 1 + (-0.906 - 0.422i)T \) |
| 13 | \( 1 + (0.906 - 0.422i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.965 - 0.258i)T \) |
| 23 | \( 1 + (0.984 - 0.173i)T \) |
| 29 | \( 1 + (0.422 - 0.906i)T \) |
| 31 | \( 1 + (0.939 + 0.342i)T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (0.342 - 0.939i)T \) |
| 43 | \( 1 + (-0.819 + 0.573i)T \) |
| 47 | \( 1 + (0.939 - 0.342i)T \) |
| 53 | \( 1 + (-0.258 + 0.965i)T \) |
| 59 | \( 1 + (-0.996 - 0.0871i)T \) |
| 61 | \( 1 + (-0.422 + 0.906i)T \) |
| 67 | \( 1 + (-0.819 - 0.573i)T \) |
| 71 | \( 1 + (0.866 - 0.5i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.422 - 0.906i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.173 - 0.984i)T \) |
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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
−18.19948440038976066108698974635, −17.4911720298290878312096965723, −16.71433437007530392490528790917, −15.877105583287795435637419430760, −15.45524775608763025679296018288, −14.92410388287906485501948914877, −14.050427911214261059889645792498, −13.568063660518021111617967061731, −12.81236309056889736755976656652, −12.083279470543997662758352365393, −11.250182532825656465846826538047, −10.89743641316172698874581209913, −10.20699657307883872514329681, −9.48906339184326945547957521598, −8.53221845107939115485028542956, −8.10396495481241435883408016591, −7.10202791817693081051139229287, −6.77214017840748823970042618021, −6.020120547214208880758887327, −5.00942887162412432825894987825, −4.437848901774607156820628979597, −3.53601996098153325950128280026, −2.85391727529232803937777382232, −2.21777094719940555100732570412, −1.174610990862223265880658340904,
0.15205509596810842077932843907, 1.0174035558155787662063775761, 1.93710101859807695005316943543, 2.86177862792448042699899035121, 3.66543936675028289641486498917, 4.436802139290523208280135547, 4.98936637010049436088652799370, 5.90064542564416128205575099484, 6.3620614801351836897457528401, 7.48630735093328148256822561205, 8.05479798957388630252287497917, 8.80628361386450925026775244025, 8.9583100185339354495523104527, 10.316014076398648602386211359915, 10.67789412479095403333085664032, 11.34661883630289710687617345011, 12.28423308008966636513311218244, 12.72456943187975619708874974447, 13.44417321176738863158253499162, 13.77360546000362386186728852620, 15.111981725726732963485407483470, 15.38741775311855417630690799994, 15.961746318781618871268530173699, 16.74222462683737891454016506178, 17.28744902879439642545071587731