L(s) = 1 | + (0.438 + 0.898i)2-s + (−0.615 + 0.788i)4-s + (0.173 + 0.984i)5-s + (0.848 − 0.529i)7-s + (−0.978 − 0.207i)8-s + (−0.809 + 0.587i)10-s + (−0.0348 − 0.999i)11-s + (−0.559 − 0.829i)13-s + (0.848 + 0.529i)14-s + (−0.241 − 0.970i)16-s + (0.978 + 0.207i)17-s + (−0.809 + 0.587i)19-s + (−0.882 − 0.469i)20-s + (0.882 − 0.469i)22-s + (−0.0348 + 0.999i)23-s + ⋯ |
L(s) = 1 | + (0.438 + 0.898i)2-s + (−0.615 + 0.788i)4-s + (0.173 + 0.984i)5-s + (0.848 − 0.529i)7-s + (−0.978 − 0.207i)8-s + (−0.809 + 0.587i)10-s + (−0.0348 − 0.999i)11-s + (−0.559 − 0.829i)13-s + (0.848 + 0.529i)14-s + (−0.241 − 0.970i)16-s + (0.978 + 0.207i)17-s + (−0.809 + 0.587i)19-s + (−0.882 − 0.469i)20-s + (0.882 − 0.469i)22-s + (−0.0348 + 0.999i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.281038253 + 1.217666775i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.281038253 + 1.217666775i\) |
\(L(1)\) |
\(\approx\) |
\(1.206605829 + 0.6567652928i\) |
\(L(1)\) |
\(\approx\) |
\(1.206605829 + 0.6567652928i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.438 + 0.898i)T \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 7 | \( 1 + (0.848 - 0.529i)T \) |
| 11 | \( 1 + (-0.0348 - 0.999i)T \) |
| 13 | \( 1 + (-0.559 - 0.829i)T \) |
| 17 | \( 1 + (0.978 + 0.207i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (-0.0348 + 0.999i)T \) |
| 29 | \( 1 + (-0.438 - 0.898i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.961 - 0.275i)T \) |
| 43 | \( 1 + (-0.438 - 0.898i)T \) |
| 47 | \( 1 + (0.961 + 0.275i)T \) |
| 53 | \( 1 + (0.978 + 0.207i)T \) |
| 59 | \( 1 + (0.559 + 0.829i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.939 - 0.342i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.978 - 0.207i)T \) |
| 79 | \( 1 + (0.615 + 0.788i)T \) |
| 83 | \( 1 + (0.997 + 0.0697i)T \) |
| 89 | \( 1 + (-0.669 - 0.743i)T \) |
| 97 | \( 1 + (0.848 - 0.529i)T \) |
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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
−21.5865083617583826027208408873, −21.0182468761023731298748629960, −20.412084892355013973256919956283, −19.643692779336813652065712611318, −18.72440048286890562120863854430, −17.95126845861529782230890913915, −17.15630883210729371092715409757, −16.25283132754626299927684751139, −14.89734628766388106454431811743, −14.64727146690383165831668754295, −13.54610254032150382634209261633, −12.602911483589608476856641183969, −12.17360611081075895271941002688, −11.40832042606750254723802284268, −10.34325977571082505495991041889, −9.450682490004602814916355835817, −8.84929892461825111208046917675, −7.86912189292099786938825494212, −6.51331420274664011247991431649, −5.3013147038881962161654817342, −4.7824840421850283678399540806, −4.09683219854374070135307822513, −2.51646727299765886014405679010, −1.86842851964621144541004367250, −0.86574241919529924379955092650,
0.62212476319682703587874668064, 2.314456699458961282083853261724, 3.44776393445989645434225369980, 4.109229177653932429460668613229, 5.57049619097114258470208018995, 5.82673005413325374039529746588, 7.15823654137156225623366193876, 7.71352060425052481397130278760, 8.42351023258811512896347230414, 9.72374689379376143276568417628, 10.60635937587300928611341375833, 11.44565261915169477040567105104, 12.44008131326638913903651000463, 13.52170968253902700961473967840, 14.08795723770466873078789010576, 14.81207024577738343100921269615, 15.354028164604996843372340078359, 16.53367915474226160791644423654, 17.20841600736679167528369710196, 17.858283657932778362356746950526, 18.707092518844655146447151677884, 19.4880214066841709373627312788, 20.8962416784780861521272182465, 21.39394041653505608395852624343, 22.15830612037221458565937615041