L(s) = 1 | + (−0.930 + 0.365i)2-s + (0.733 − 0.680i)4-s + (−0.433 + 0.900i)8-s + (0.623 − 0.781i)11-s + (−0.930 + 0.365i)13-s + (0.0747 − 0.997i)16-s + (−0.680 + 0.733i)17-s + (−0.5 − 0.866i)19-s + (−0.294 + 0.955i)22-s + (−0.974 + 0.222i)23-s + (0.733 − 0.680i)26-s + (−0.955 + 0.294i)29-s + (0.5 + 0.866i)31-s + (0.294 + 0.955i)32-s + (0.365 − 0.930i)34-s + ⋯ |
L(s) = 1 | + (−0.930 + 0.365i)2-s + (0.733 − 0.680i)4-s + (−0.433 + 0.900i)8-s + (0.623 − 0.781i)11-s + (−0.930 + 0.365i)13-s + (0.0747 − 0.997i)16-s + (−0.680 + 0.733i)17-s + (−0.5 − 0.866i)19-s + (−0.294 + 0.955i)22-s + (−0.974 + 0.222i)23-s + (0.733 − 0.680i)26-s + (−0.955 + 0.294i)29-s + (0.5 + 0.866i)31-s + (0.294 + 0.955i)32-s + (0.365 − 0.930i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7194471567 - 0.2157388760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7194471567 - 0.2157388760i\) |
\(L(1)\) |
\(\approx\) |
\(0.6506390747 + 0.03561977760i\) |
\(L(1)\) |
\(\approx\) |
\(0.6506390747 + 0.03561977760i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.930 + 0.365i)T \) |
| 11 | \( 1 + (0.623 - 0.781i)T \) |
| 13 | \( 1 + (-0.930 + 0.365i)T \) |
| 17 | \( 1 + (-0.680 + 0.733i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.974 + 0.222i)T \) |
| 29 | \( 1 + (-0.955 + 0.294i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.294 + 0.955i)T \) |
| 41 | \( 1 + (-0.826 + 0.563i)T \) |
| 43 | \( 1 + (0.563 - 0.826i)T \) |
| 47 | \( 1 + (0.930 - 0.365i)T \) |
| 53 | \( 1 + (0.294 - 0.955i)T \) |
| 59 | \( 1 + (0.826 + 0.563i)T \) |
| 61 | \( 1 + (0.733 + 0.680i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.222 - 0.974i)T \) |
| 73 | \( 1 + (-0.149 - 0.988i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.930 + 0.365i)T \) |
| 89 | \( 1 + (0.365 - 0.930i)T \) |
| 97 | \( 1 + (-0.866 + 0.5i)T \) |
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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
−19.84563824313427180292845761876, −19.02660063298284086089929580580, −18.42680864798198816096898777583, −17.51635400356135782970162624456, −17.21299278199030904543372586766, −16.36386949903232748025694259687, −15.589509012370137220212984811876, −14.87370577451000805714913803206, −14.09219398907460166840954712181, −12.92686020493260478417221301817, −12.37148606261091657312538415080, −11.69343744352941365874467868075, −10.963634911319768906321016323124, −9.9696650854410839903379999065, −9.658950602564829512839879492719, −8.80196354416854903569732756428, −7.89316258439947078175321480340, −7.31554469151610216859441128302, −6.534787218176741077957333157974, −5.621382855865933672886600399610, −4.35782916024592389109870411527, −3.76202041177828288501961303316, −2.39050663232848782777817032611, −2.11126916902842510318602987273, −0.782797044207750857347014225563,
0.45182617483447069103355121561, 1.683887696637158121887798187078, 2.38945433097797716766432205697, 3.52648563732397672064364670283, 4.59648322164333037478487338560, 5.526818577589516255228242839340, 6.42659309133845003336131655724, 6.90382000137661491484766426761, 7.84783151116957949461572474596, 8.67173086922319805697103849139, 9.12135376263320640012164296761, 10.05600436665980939772835942356, 10.69257935869605703578623143231, 11.55072691686434487413019810959, 12.05537450060409821106367959061, 13.23219846369881328236120794915, 14.02212034851384725860917539138, 14.83417809007358029635176800550, 15.3520035356273371850192225953, 16.262890418720576060247159242374, 16.87784987384151719644466384526, 17.435580315492233396919939954814, 18.130600975843960024918122956277, 19.05729349462414319726715153923, 19.5147474238250936074212759777