L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + 7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s − 18-s + (−0.309 − 0.951i)21-s + (−0.309 − 0.951i)22-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + 7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s − 18-s + (−0.309 − 0.951i)21-s + (−0.309 − 0.951i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.382213936 - 1.309633194i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.382213936 - 1.309633194i\) |
\(L(1)\) |
\(\approx\) |
\(1.562000044 - 0.09827271304i\) |
\(L(1)\) |
\(\approx\) |
\(1.562000044 - 0.09827271304i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.809 - 0.587i)T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.809 - 0.587i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + (-0.309 + 0.951i)T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.309 - 0.951i)T \) |
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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
−23.65765073494971490069889177682, −22.85130206193749783912156299147, −21.7697999069437992724396080009, −21.33583120546365815600848685068, −20.58678047101553881510829309908, −19.96401356488614128527221454378, −18.61082518221396346470479981159, −17.86293153732406559315820821720, −16.69675302168634205586181991861, −15.73792507864737970143387741057, −14.99613766105090790036820166775, −14.34721275498256305434832486758, −13.3227925450226960415301406648, −12.23767742323704531366387285177, −11.35950727961730788569483227131, −10.75234907138808571228911321415, −9.94461400952548313513926780616, −8.89154600098289216497632834487, −7.64264514053143736299322519795, −6.10675003724037026459703018275, −5.41755850142900032064871541644, −4.412500304871735401232179295444, −3.8170735351380742220187626507, −2.44798078502029631672818545814, −1.27756983396693740410237665064,
0.564732304760017636561915565025, 2.08456493679202601962358998546, 3.083892463198982602325693371800, 4.52231249272359272227519974011, 5.55127283695531284188922896392, 6.074271361435874899630330210026, 7.43334992311980320399932119960, 7.90835444860019480281094491416, 8.75103692445167290454972409727, 10.74216221426197827448896785560, 11.37914831592754625644559081475, 12.2940145835368847624685999509, 13.1423025573747630526208525881, 13.91951833619876493298109517406, 14.525600342738477975376296537064, 15.819555792731951423928354121648, 16.41279684285965479360016516011, 17.714850197677633261783208729585, 17.96615830627296101776971923425, 19.0200662933830947429375151200, 20.44463912300733813775173655542, 20.8937368838309154616930058920, 22.00472401570690172222443546889, 22.914967387428380683905915325, 23.541582181242197705123391603150