L(s) = 1 | + (−0.766 − 0.642i)3-s + (0.939 − 0.342i)5-s + (−0.5 − 0.866i)7-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (−0.173 + 0.984i)21-s + (−0.939 − 0.342i)23-s + (0.766 − 0.642i)25-s + (0.5 − 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.939 + 0.342i)33-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)3-s + (0.939 − 0.342i)5-s + (−0.5 − 0.866i)7-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (−0.173 + 0.984i)21-s + (−0.939 − 0.342i)23-s + (0.766 − 0.642i)25-s + (0.5 − 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.939 + 0.342i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5546247993 - 0.6712391888i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5546247993 - 0.6712391888i\) |
\(L(1)\) |
\(\approx\) |
\(0.7977802953 - 0.3878781122i\) |
\(L(1)\) |
\(\approx\) |
\(0.7977802953 - 0.3878781122i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 5 | \( 1 + (0.939 - 0.342i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.766 + 0.642i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.173 - 0.984i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.939 - 0.342i)T \) |
| 47 | \( 1 + (0.173 + 0.984i)T \) |
| 53 | \( 1 + (0.939 + 0.342i)T \) |
| 59 | \( 1 + (-0.173 + 0.984i)T \) |
| 61 | \( 1 + (0.939 + 0.342i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.939 + 0.342i)T \) |
| 73 | \( 1 + (0.766 + 0.642i)T \) |
| 79 | \( 1 + (0.766 + 0.642i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.173 - 0.984i)T \) |
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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
−28.20611790958829530871539883978, −27.63587015378261283240740339099, −26.23363653363231728138825613760, −25.53362441037814091151842698, −24.49089950479604570453361306782, −23.12968429467007709441283070676, −22.04253416829264833765301299837, −21.919727632276209430613495182749, −20.62641651391601476194032962552, −19.36725642974778087124089660428, −17.945150958792768617306840228192, −17.512689907585373611278239676762, −16.34189043323290356311112552343, −15.20052826553794468606563999343, −14.46612754270219564956381227898, −12.73839791558192166162559565653, −12.1178571410542125217784041445, −10.59090077246945154683585887653, −9.87285866445738178245054919007, −8.96633241650142575608887463632, −6.99877147699615075330693097579, −5.93202846309373085223735776689, −5.13964671642916839912680042555, −3.526751898540040130291726725, −1.97495632397080903605018970567,
0.83595704519343706261788627905, 2.34881049249478035831683816746, 4.302519759197259301828826653495, 5.662826318568666248229703819387, 6.54172291463389107039495880384, 7.60815863476138183383663690578, 9.25468995976436701894438247147, 10.2385991730668460600359921896, 11.44274257716122545896274532059, 12.497499766323358026197702876317, 13.622864332286052368534353371464, 14.11030777682482614086006293413, 16.250998196212757804589041812376, 16.78853837207329485066037237375, 17.62164150155039232806067466488, 18.75957090822318265557700779429, 19.69183664775121147536557814673, 20.940110709201329220576663478721, 22.09785598641606574324180920382, 22.735779724273175143929517591950, 24.12003173451698370867538166498, 24.51219110687196804359929389642, 25.72940262989066845782320630634, 26.78889592074783653570079928326, 27.945446122901085076673855558294