L(s) = 1 | + (0.460 − 0.887i)2-s + (−0.334 − 0.942i)3-s + (−0.576 − 0.816i)4-s + (0.917 − 0.398i)5-s + (−0.990 − 0.136i)6-s + (0.854 − 0.519i)7-s + (−0.990 + 0.136i)8-s + (−0.775 + 0.631i)9-s + (0.0682 − 0.997i)10-s + (−0.962 − 0.269i)11-s + (−0.576 + 0.816i)12-s + (−0.203 + 0.979i)13-s + (−0.0682 − 0.997i)14-s + (−0.682 − 0.730i)15-s + (−0.334 + 0.942i)16-s + (0.962 − 0.269i)17-s + ⋯ |
L(s) = 1 | + (0.460 − 0.887i)2-s + (−0.334 − 0.942i)3-s + (−0.576 − 0.816i)4-s + (0.917 − 0.398i)5-s + (−0.990 − 0.136i)6-s + (0.854 − 0.519i)7-s + (−0.990 + 0.136i)8-s + (−0.775 + 0.631i)9-s + (0.0682 − 0.997i)10-s + (−0.962 − 0.269i)11-s + (−0.576 + 0.816i)12-s + (−0.203 + 0.979i)13-s + (−0.0682 − 0.997i)14-s + (−0.682 − 0.730i)15-s + (−0.334 + 0.942i)16-s + (0.962 − 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2706791052 - 1.758269982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2706791052 - 1.758269982i\) |
\(L(1)\) |
\(\approx\) |
\(0.7896398626 - 1.069991554i\) |
\(L(1)\) |
\(\approx\) |
\(0.7896398626 - 1.069991554i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 47 | \( 1 \) |
good | 2 | \( 1 + (0.460 - 0.887i)T \) |
| 3 | \( 1 + (-0.334 - 0.942i)T \) |
| 5 | \( 1 + (0.917 - 0.398i)T \) |
| 7 | \( 1 + (0.854 - 0.519i)T \) |
| 11 | \( 1 + (-0.962 - 0.269i)T \) |
| 13 | \( 1 + (-0.203 + 0.979i)T \) |
| 17 | \( 1 + (0.962 - 0.269i)T \) |
| 19 | \( 1 + (0.917 + 0.398i)T \) |
| 23 | \( 1 + (-0.460 - 0.887i)T \) |
| 29 | \( 1 + (-0.203 - 0.979i)T \) |
| 31 | \( 1 + (0.334 - 0.942i)T \) |
| 37 | \( 1 + (-0.0682 + 0.997i)T \) |
| 41 | \( 1 + (0.990 + 0.136i)T \) |
| 43 | \( 1 + (0.576 + 0.816i)T \) |
| 53 | \( 1 + (-0.990 - 0.136i)T \) |
| 59 | \( 1 + (-0.576 + 0.816i)T \) |
| 61 | \( 1 + (-0.0682 - 0.997i)T \) |
| 67 | \( 1 + (-0.854 - 0.519i)T \) |
| 71 | \( 1 + (0.460 + 0.887i)T \) |
| 73 | \( 1 + (0.775 + 0.631i)T \) |
| 79 | \( 1 + (0.682 + 0.730i)T \) |
| 83 | \( 1 + (0.962 + 0.269i)T \) |
| 89 | \( 1 + (-0.917 + 0.398i)T \) |
| 97 | \( 1 + (-0.334 - 0.942i)T \) |
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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
−34.09607741204914068515200288150, −33.30315335907702788283593346768, −32.31278303284865489416725029337, −31.200445762779245598668355584433, −29.86777981983487475778328307222, −28.28785735888778695750482559608, −27.159084512168229664515452112883, −26.02589080179972908948398110184, −25.13668867820734322551124206691, −23.69344245599365045930253478733, −22.46697249177646859087182784797, −21.54883922040776086320218315077, −20.76788210532382771164769895182, −18.03161326237663048457698311116, −17.5522642603088931559536197379, −16.00405472766774929548874167291, −14.99799963604229621761968738971, −14.02363224187530867765495551362, −12.3664204372937298254441690132, −10.63789844165464315009554912533, −9.28479106532600600658403404588, −7.69358768272395467704164667573, −5.670131189158640930047663469043, −5.14197486147715831827556893964, −3.07331567344334262029658302349,
1.05013691721279473794140043946, 2.36815815181982542814764062125, 4.82165329179652783949427329306, 5.99619741377964519895991173558, 7.99139893671675170666116811504, 9.81760610585220463658739876774, 11.20065942260307638739382444083, 12.36532962729688931052041155431, 13.62834440040961393327437017386, 14.23099770996834251882774454100, 16.74372036479254229986851832941, 18.01471175151146958004806430073, 18.83560753992587369031414419231, 20.468520510080829860476536649, 21.23138339580834041028425722901, 22.68469064837813998098683301613, 23.92303995463623569833302658023, 24.54449661367772181168982044048, 26.36191286881432928438923979919, 28.03679133564430085656082042166, 28.97031397642627463757676408420, 29.71213439769697087098797774121, 30.755607767720296258384101319239, 31.80948155476705484726593048982, 33.296660725072443682836556870155