L(s) = 1 | + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)5-s + (0.900 + 0.433i)7-s + (0.623 + 0.781i)9-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)15-s − 17-s + (−0.900 + 0.433i)19-s + (−0.623 − 0.781i)21-s + (0.222 − 0.974i)23-s + (−0.900 + 0.433i)25-s + (−0.222 − 0.974i)27-s + (−0.222 − 0.974i)31-s + (−0.900 + 0.433i)33-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)5-s + (0.900 + 0.433i)7-s + (0.623 + 0.781i)9-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)15-s − 17-s + (−0.900 + 0.433i)19-s + (−0.623 − 0.781i)21-s + (0.222 − 0.974i)23-s + (−0.900 + 0.433i)25-s + (−0.222 − 0.974i)27-s + (−0.222 − 0.974i)31-s + (−0.900 + 0.433i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3700382744 - 0.9309512216i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3700382744 - 0.9309512216i\) |
\(L(1)\) |
\(\approx\) |
\(0.7418966932 - 0.3609039609i\) |
\(L(1)\) |
\(\approx\) |
\(0.7418966932 - 0.3609039609i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (-0.900 - 0.433i)T \) |
| 5 | \( 1 + (-0.222 - 0.974i)T \) |
| 7 | \( 1 + (0.900 + 0.433i)T \) |
| 11 | \( 1 + (0.623 - 0.781i)T \) |
| 13 | \( 1 + (0.623 - 0.781i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (-0.900 + 0.433i)T \) |
| 23 | \( 1 + (0.222 - 0.974i)T \) |
| 31 | \( 1 + (-0.222 - 0.974i)T \) |
| 37 | \( 1 + (-0.623 - 0.781i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (0.623 - 0.781i)T \) |
| 53 | \( 1 + (-0.222 - 0.974i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (0.900 + 0.433i)T \) |
| 67 | \( 1 + (-0.623 - 0.781i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.222 - 0.974i)T \) |
| 79 | \( 1 + (0.623 + 0.781i)T \) |
| 83 | \( 1 + (0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.222 + 0.974i)T \) |
| 97 | \( 1 + (0.900 - 0.433i)T \) |
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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
−29.39630236879131770500742869289, −28.21533447540567617889228326195, −27.35437704281223397725874832375, −26.612027842415981944958586460360, −25.48163918126750543435362149264, −23.84664692861031135161498832201, −23.338822602805868809382400450679, −22.19898880395788810549936443229, −21.45044519415071394803074774053, −20.23231946800144910196048349696, −18.86557639296722033663144860108, −17.72557048074977710961937979413, −17.17440483303410130404001597469, −15.661432922315774937483899478924, −14.88095297698813281786551597013, −13.68422564502889929538093955242, −11.97599408588127071195967049795, −11.16289680699565123375085437958, −10.41617285761140462749461685622, −8.967345198660467885533812466486, −7.18585860245597687574030948736, −6.4438696631553574131692179394, −4.766657294874706325755261119307, −3.82644700916630757462806847640, −1.68532949907392948284878133972,
0.48540657349548177258942072483, 1.81487301258962716689771888051, 4.202510078246022221902240202991, 5.33099020864045719247001942250, 6.35479402886231889162471954369, 8.036574577978951176872075268760, 8.81528334635240403417324309182, 10.71430556869625067727891474723, 11.56303213509923316826855597517, 12.566069643734215363437941179245, 13.51842914359371765458459814154, 15.10505341486239895871098366690, 16.302467353683596203040259750006, 17.153063569077814418186798308250, 18.09370690162167362187760838060, 19.17780931777788366158382801959, 20.464194851215106279630863634219, 21.477956107266114986841466990129, 22.553489342757701342313712346565, 23.69825931996011584650989854920, 24.48411453865095152537222093334, 25.06798227099674184757575058805, 27.00047639661863025054459155608, 27.80806015986273436316260926185, 28.42557725885463429179096341464