L(s) = 1 | + (0.861 − 0.508i)2-s + (0.669 − 0.743i)3-s + (0.483 − 0.875i)4-s + (−0.761 + 0.647i)5-s + (0.198 − 0.980i)6-s + (−0.0285 − 0.999i)8-s + (−0.104 − 0.994i)9-s + (−0.327 + 0.945i)10-s + (−0.327 − 0.945i)12-s + (0.0855 − 0.996i)13-s + (−0.0285 + 0.999i)15-s + (−0.532 − 0.846i)16-s + (0.449 + 0.893i)17-s + (−0.595 − 0.803i)18-s + (−0.710 − 0.703i)19-s + (0.198 + 0.980i)20-s + ⋯ |
L(s) = 1 | + (0.861 − 0.508i)2-s + (0.669 − 0.743i)3-s + (0.483 − 0.875i)4-s + (−0.761 + 0.647i)5-s + (0.198 − 0.980i)6-s + (−0.0285 − 0.999i)8-s + (−0.104 − 0.994i)9-s + (−0.327 + 0.945i)10-s + (−0.327 − 0.945i)12-s + (0.0855 − 0.996i)13-s + (−0.0285 + 0.999i)15-s + (−0.532 − 0.846i)16-s + (0.449 + 0.893i)17-s + (−0.595 − 0.803i)18-s + (−0.710 − 0.703i)19-s + (0.198 + 0.980i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8305147378 - 2.309438808i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8305147378 - 2.309438808i\) |
\(L(1)\) |
\(\approx\) |
\(1.389191676 - 1.130038604i\) |
\(L(1)\) |
\(\approx\) |
\(1.389191676 - 1.130038604i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.861 - 0.508i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (-0.761 + 0.647i)T \) |
| 13 | \( 1 + (0.0855 - 0.996i)T \) |
| 17 | \( 1 + (0.449 + 0.893i)T \) |
| 19 | \( 1 + (-0.710 - 0.703i)T \) |
| 23 | \( 1 + (0.928 + 0.371i)T \) |
| 29 | \( 1 + (0.774 - 0.633i)T \) |
| 31 | \( 1 + (-0.290 - 0.956i)T \) |
| 37 | \( 1 + (0.123 - 0.992i)T \) |
| 41 | \( 1 + (-0.736 + 0.676i)T \) |
| 43 | \( 1 + (-0.959 - 0.281i)T \) |
| 47 | \( 1 + (-0.595 + 0.803i)T \) |
| 53 | \( 1 + (-0.532 + 0.846i)T \) |
| 59 | \( 1 + (-0.217 + 0.976i)T \) |
| 61 | \( 1 + (0.00951 - 0.999i)T \) |
| 67 | \( 1 + (0.580 - 0.814i)T \) |
| 71 | \( 1 + (-0.362 - 0.931i)T \) |
| 73 | \( 1 + (0.345 + 0.938i)T \) |
| 79 | \( 1 + (0.380 + 0.924i)T \) |
| 83 | \( 1 + (0.696 - 0.717i)T \) |
| 89 | \( 1 + (0.0475 + 0.998i)T \) |
| 97 | \( 1 + (0.941 - 0.336i)T \) |
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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
−22.43724462453555282926478658220, −21.515328232051802595012775720704, −20.926612633363361144580606282364, −20.339166089873725048655552654415, −19.4856644777482786751874251268, −18.62692681176673471812266455476, −17.10593911748815496585434696941, −16.40036203124140192206607182509, −16.06173513312741784742872384300, −15.08239054707361827537623670681, −14.4948336323566133485781776175, −13.70219356160543350905761062105, −12.82617749598589805696888496969, −11.93467185233053567100282820850, −11.22456938524851636474975863943, −10.08660327897848792996812241987, −8.81040386685444684195686694833, −8.47147544540717070673351627215, −7.42671544513378578639525675747, −6.58475354109645281876175253381, −5.04581397739830998637101066292, −4.78734009755399668109989248560, −3.72450948277742360282159200444, −3.10385743896823714668782811174, −1.77472805635873335614093881153,
0.74441156386142306131099886649, 2.03891756852137240478608644950, 2.99241956176786802082851902974, 3.55043360022137328816977156681, 4.59118452666831371114773247408, 5.92851062035702861932668458365, 6.65279348285204604191582850199, 7.5709571514566703151056392005, 8.32321997681979745693500909029, 9.560280167221036060442591679280, 10.60203623518867440999464854744, 11.28833125739613640777612252619, 12.2343773611300785570605835508, 12.88325601305515250022313235783, 13.57193897846804168170950051468, 14.578975829915260627712498745660, 15.11806780602301308118789807698, 15.60432817499503367951819065213, 17.092167902300240664412177590306, 18.17846365689918129674038352082, 18.91794750431086553787298223668, 19.56663950807069945203039952654, 20.01857117396222086487930454375, 21.00144955868680666205171966828, 21.74945706360448068303466295241