L(s) = 1 | + (−0.181 + 0.983i)2-s + (−0.322 + 0.946i)3-s + (−0.934 − 0.357i)4-s + (−0.581 + 0.813i)5-s + (−0.872 − 0.489i)6-s + (0.905 + 0.424i)7-s + (0.520 − 0.853i)8-s + (−0.791 − 0.611i)9-s + (−0.694 − 0.719i)10-s + (0.252 − 0.967i)11-s + (0.639 − 0.768i)12-s + (−0.872 + 0.489i)13-s + (−0.581 + 0.813i)14-s + (−0.581 − 0.813i)15-s + (0.744 + 0.667i)16-s + (0.957 − 0.288i)17-s + ⋯ |
L(s) = 1 | + (−0.181 + 0.983i)2-s + (−0.322 + 0.946i)3-s + (−0.934 − 0.357i)4-s + (−0.581 + 0.813i)5-s + (−0.872 − 0.489i)6-s + (0.905 + 0.424i)7-s + (0.520 − 0.853i)8-s + (−0.791 − 0.611i)9-s + (−0.694 − 0.719i)10-s + (0.252 − 0.967i)11-s + (0.639 − 0.768i)12-s + (−0.872 + 0.489i)13-s + (−0.581 + 0.813i)14-s + (−0.581 − 0.813i)15-s + (0.744 + 0.667i)16-s + (0.957 − 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3385433506 + 0.7019026389i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3385433506 + 0.7019026389i\) |
\(L(1)\) |
\(\approx\) |
\(0.4190233193 + 0.6130142142i\) |
\(L(1)\) |
\(\approx\) |
\(0.4190233193 + 0.6130142142i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (-0.181 + 0.983i)T \) |
| 3 | \( 1 + (-0.322 + 0.946i)T \) |
| 5 | \( 1 + (-0.581 + 0.813i)T \) |
| 7 | \( 1 + (0.905 + 0.424i)T \) |
| 11 | \( 1 + (0.252 - 0.967i)T \) |
| 13 | \( 1 + (-0.872 + 0.489i)T \) |
| 17 | \( 1 + (0.957 - 0.288i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.639 + 0.768i)T \) |
| 29 | \( 1 + (-0.791 + 0.611i)T \) |
| 31 | \( 1 + (-0.872 - 0.489i)T \) |
| 37 | \( 1 + (0.989 + 0.145i)T \) |
| 41 | \( 1 + (-0.791 - 0.611i)T \) |
| 47 | \( 1 + (-0.791 + 0.611i)T \) |
| 53 | \( 1 + (0.520 + 0.853i)T \) |
| 59 | \( 1 + (-0.457 + 0.889i)T \) |
| 61 | \( 1 + (0.905 + 0.424i)T \) |
| 67 | \( 1 + (-0.181 + 0.983i)T \) |
| 71 | \( 1 + (-0.791 - 0.611i)T \) |
| 73 | \( 1 + (-0.934 + 0.357i)T \) |
| 79 | \( 1 + (-0.0365 + 0.999i)T \) |
| 83 | \( 1 + (0.905 + 0.424i)T \) |
| 89 | \( 1 + (-0.976 - 0.217i)T \) |
| 97 | \( 1 + (-0.872 - 0.489i)T \) |
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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
−19.812210991533529508814824841905, −19.00494102446204180275239504793, −18.2254169842418524189458787105, −17.60440249188899082454446446108, −16.96152540422243045297374112120, −16.46765995893561234444474038990, −14.81053089929796785232927764789, −14.5046221972554313298517476611, −13.28836659292127010475869626800, −12.85842016728706069233044686376, −12.01577482841449018049949019428, −11.760018119222131029788449301557, −10.879325649565555471284803130031, −9.96936816653703077570478646972, −9.1427894503876500495742488074, −8.04073868654323031926047081999, −7.80678692748825838479664279884, −7.00898212245433082657999632499, −5.31626033060218787037764618103, −5.0507831360063379212553210430, −4.093059551510199146253821856090, −3.04188804392898741019523840267, −1.88497052865021363672002690265, −1.31961897414193142338119961117, −0.36793600552186067042284619945,
1.15837187093184357214188759841, 2.88299023720690189072746132030, 3.67596024198566408691494835034, 4.51352328295658062390213658333, 5.4541365431236517051448225103, 5.775811493724790947628375378653, 7.06253750021050308496208303150, 7.59261810336087737982931949547, 8.49967326003785611144465605679, 9.28857959387266551003242142447, 9.91822596353918913270765460639, 10.94653830604901357340760842569, 11.473043862651713588191799237716, 12.18190794251873302045652641307, 13.65591712514576720572216619662, 14.41799945769100044498485515706, 14.783760010361044760503451599502, 15.37243040702134164155357287855, 16.3258584833145670466657114608, 16.657011815244870311046709335516, 17.566634256806678343259685809359, 18.31465270619605127558425239152, 18.89178986843935821666311603707, 19.70516000461408620297131975971, 20.73614391505010944760873311259