L(s) = 1 | + (−0.104 − 0.994i)3-s + (−0.669 + 0.743i)5-s + (−0.978 + 0.207i)9-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)15-s + (−0.978 − 0.207i)17-s + (−0.913 − 0.406i)19-s + (0.5 + 0.866i)23-s + (−0.104 − 0.994i)25-s + (0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (0.669 + 0.743i)31-s + (−0.104 + 0.994i)37-s + (−0.978 − 0.207i)39-s + (−0.809 + 0.587i)41-s + ⋯ |
L(s) = 1 | + (−0.104 − 0.994i)3-s + (−0.669 + 0.743i)5-s + (−0.978 + 0.207i)9-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)15-s + (−0.978 − 0.207i)17-s + (−0.913 − 0.406i)19-s + (0.5 + 0.866i)23-s + (−0.104 − 0.994i)25-s + (0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (0.669 + 0.743i)31-s + (−0.104 + 0.994i)37-s + (−0.978 − 0.207i)39-s + (−0.809 + 0.587i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.173736614 + 0.1516264648i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.173736614 + 0.1516264648i\) |
\(L(1)\) |
\(\approx\) |
\(0.8549870094 - 0.1174604991i\) |
\(L(1)\) |
\(\approx\) |
\(0.8549870094 - 0.1174604991i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.669 + 0.743i)T \) |
| 13 | \( 1 + (0.309 - 0.951i)T \) |
| 17 | \( 1 + (-0.978 - 0.207i)T \) |
| 19 | \( 1 + (-0.913 - 0.406i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.104 + 0.994i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.913 + 0.406i)T \) |
| 53 | \( 1 + (0.669 + 0.743i)T \) |
| 59 | \( 1 + (0.913 - 0.406i)T \) |
| 61 | \( 1 + (0.669 - 0.743i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.913 - 0.406i)T \) |
| 79 | \( 1 + (-0.978 + 0.207i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.309 + 0.951i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.00797457112809672491126991553, −24.01169482944553224175878730075, −23.19124169444774910862148426092, −22.3555624850877118284448490426, −21.21319889901315458552506918575, −20.75535449975102476749277759922, −19.71305487228348664313292504993, −18.94041572105035009488814846288, −17.432515143082127753764499606495, −16.72174263223974379082215089023, −15.906890312661518079188212536576, −15.19945062605584793661122654446, −14.164496940813797952838837724284, −12.9602063988157353786451667735, −11.86921899954141319286600877195, −11.10534681572903117871100454240, −10.07906335836263365704937622825, −8.830673929242865047633279073508, −8.47670285137639995740973430016, −6.85270027139130908097189710233, −5.644675602423639076928591533762, −4.340477857534312025101976002301, −4.05599470993912638979117003654, −2.35622530001170693386109258764, −0.49122833555108860228049143078,
0.84625981018413152903741303287, 2.41999288428137823559611883112, 3.3499487562073911999141118727, 4.87926468857506172451566248963, 6.27603552539651900695238352691, 6.9701477870234586869237864638, 7.96316160345822720943881719877, 8.79335684254796321733295225348, 10.48522835819180508998625882994, 11.20115163172247238746704933546, 12.14287748470081877903779531343, 13.10444461081369287044478487142, 13.95458745834631097433934090668, 15.07919135274187565952850776273, 15.790061618656982047103463970175, 17.26782405799628082320227789284, 17.89987198430238253414420953543, 18.80902509388149279780245815327, 19.57577719531581970652264568501, 20.263740006681162194284427674773, 21.725280269013837866805638076407, 22.65355785534871104370227256454, 23.34460129618561336546796494198, 24.03349547021842360105959835936, 25.17345605792673819550352600906