L(s) = 1 | + (−0.725 + 0.687i)2-s + (0.0541 − 0.998i)4-s + (0.161 − 0.986i)5-s + (0.468 − 0.883i)7-s + (0.647 + 0.762i)8-s + (0.561 + 0.827i)10-s + (−0.994 + 0.108i)11-s + (−0.907 − 0.419i)13-s + (0.267 + 0.963i)14-s + (−0.994 − 0.108i)16-s + (−0.468 − 0.883i)17-s + (−0.856 + 0.515i)19-s + (−0.976 − 0.214i)20-s + (0.647 − 0.762i)22-s + (−0.370 − 0.928i)23-s + ⋯ |
L(s) = 1 | + (−0.725 + 0.687i)2-s + (0.0541 − 0.998i)4-s + (0.161 − 0.986i)5-s + (0.468 − 0.883i)7-s + (0.647 + 0.762i)8-s + (0.561 + 0.827i)10-s + (−0.994 + 0.108i)11-s + (−0.907 − 0.419i)13-s + (0.267 + 0.963i)14-s + (−0.994 − 0.108i)16-s + (−0.468 − 0.883i)17-s + (−0.856 + 0.515i)19-s + (−0.976 − 0.214i)20-s + (0.647 − 0.762i)22-s + (−0.370 − 0.928i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0900 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0900 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4424890225 - 0.4042664035i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4424890225 - 0.4042664035i\) |
\(L(1)\) |
\(\approx\) |
\(0.6602676099 - 0.1150557210i\) |
\(L(1)\) |
\(\approx\) |
\(0.6602676099 - 0.1150557210i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 \) |
good | 2 | \( 1 + (-0.725 + 0.687i)T \) |
| 5 | \( 1 + (0.161 - 0.986i)T \) |
| 7 | \( 1 + (0.468 - 0.883i)T \) |
| 11 | \( 1 + (-0.994 + 0.108i)T \) |
| 13 | \( 1 + (-0.907 - 0.419i)T \) |
| 17 | \( 1 + (-0.468 - 0.883i)T \) |
| 19 | \( 1 + (-0.856 + 0.515i)T \) |
| 23 | \( 1 + (-0.370 - 0.928i)T \) |
| 29 | \( 1 + (0.725 + 0.687i)T \) |
| 31 | \( 1 + (0.856 + 0.515i)T \) |
| 37 | \( 1 + (-0.647 + 0.762i)T \) |
| 41 | \( 1 + (0.370 - 0.928i)T \) |
| 43 | \( 1 + (0.994 + 0.108i)T \) |
| 47 | \( 1 + (-0.161 - 0.986i)T \) |
| 53 | \( 1 + (0.561 - 0.827i)T \) |
| 61 | \( 1 + (0.725 - 0.687i)T \) |
| 67 | \( 1 + (-0.647 - 0.762i)T \) |
| 71 | \( 1 + (0.161 + 0.986i)T \) |
| 73 | \( 1 + (-0.267 - 0.963i)T \) |
| 79 | \( 1 + (0.976 + 0.214i)T \) |
| 83 | \( 1 + (0.796 + 0.605i)T \) |
| 89 | \( 1 + (-0.725 - 0.687i)T \) |
| 97 | \( 1 + (-0.267 + 0.963i)T \) |
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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
−27.70208981891333110040721488733, −26.57992963804120942563572862174, −26.07836220953793687289223797578, −25.0334027871045128826494250863, −23.82698372125548646744628039866, −22.444144582089253942288810406197, −21.54045839493527399186485963777, −21.14559335971058525265785321876, −19.44914376875205830995881048124, −19.042804503084710960572397755359, −17.86275570937979041654245270735, −17.418499654175643212064563353941, −15.79140215869181815581740504835, −14.95290886471633603198293893504, −13.59137526168742502913829542901, −12.415438778184615717848756144473, −11.392031687669186243961754660484, −10.56273292574565603227946621686, −9.562379995842849383244315628989, −8.35305085712192752013027542776, −7.39889720709378121835781259614, −6.04778148028645691986867154587, −4.38048159373769942805209434068, −2.7351062262919998180468114537, −2.10744049349503933535693607692,
0.58396569070809752316348865267, 2.17267688548215719986566484120, 4.57691561896054360363167484672, 5.23974504961773009491314984538, 6.77635571298367093145378347100, 7.87807821918139234213282985118, 8.62417628265474064728005354134, 9.97386014919092208607178276885, 10.64301916665018409988921131478, 12.225432827268591832231761161478, 13.46065132430956071074602135439, 14.40431500721909071210914882446, 15.63993603811110785831432064083, 16.49551608432068681531197955889, 17.355359617580183323560163320705, 18.064791376861904560529421507555, 19.41506194610174603974564182719, 20.32634890321100451751811873660, 20.99409016560830306042073547336, 22.71227237162385681886718327961, 23.69793892319229836198779281383, 24.3643787292745933330642377267, 25.19372304422040461073259000004, 26.29842402073313047656830209844, 27.14677988703261266393162387644