| L(s) = 1 | + (−0.104 + 0.994i)2-s + (−1.69 + 0.364i)3-s + (−0.978 − 0.207i)4-s − 1.10·5-s + (−0.185 − 1.72i)6-s + (−0.801 − 2.46i)7-s + (0.309 − 0.951i)8-s + (2.73 − 1.23i)9-s + (0.115 − 1.09i)10-s + (1.42 + 0.302i)11-s + (1.73 − 0.00414i)12-s + (−3.46 + 2.51i)13-s + (2.53 − 0.539i)14-s + (1.86 − 0.400i)15-s + (0.913 + 0.406i)16-s + (3.66 + 4.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (−0.977 + 0.210i)3-s + (−0.489 − 0.103i)4-s − 0.492·5-s + (−0.0755 − 0.703i)6-s + (−0.302 − 0.932i)7-s + (0.109 − 0.336i)8-s + (0.911 − 0.411i)9-s + (0.0363 − 0.346i)10-s + (0.428 + 0.0911i)11-s + (0.499 − 0.00119i)12-s + (−0.960 + 0.697i)13-s + (0.677 − 0.144i)14-s + (0.481 − 0.103i)15-s + (0.228 + 0.101i)16-s + (0.889 + 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.687235 + 0.451770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.687235 + 0.451770i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (1.69 - 0.364i)T \) |
| 31 | \( 1 + (-1.94 - 5.21i)T \) |
| good | 5 | \( 1 + 1.10T + 5T^{2} \) |
| 7 | \( 1 + (0.801 + 2.46i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (-1.42 - 0.302i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (3.46 - 2.51i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.66 - 4.07i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-7.18 + 3.19i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-2.73 - 3.03i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.802 + 7.63i)T + (-28.3 - 6.02i)T^{2} \) |
| 37 | \( 1 + (3.99 - 6.91i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.55 - 4.76i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (1.22 + 0.890i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-12.1 - 5.39i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (1.52 - 0.323i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-3.25 - 1.44i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (5.28 + 9.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + (-6.86 + 1.45i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-6.54 + 7.27i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (4.27 - 13.1i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-8.37 + 3.72i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (0.226 + 0.696i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.44 + 2.71i)T + (-10.1 - 96.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.92911780702903264442361009936, −9.829971290826598768091996140023, −9.512194393586177915052489589417, −7.900657675468575489517370841037, −7.23217953831110138328297487839, −6.51871745488354611689631748557, −5.40854623844774600384095596170, −4.47701107794998510770500510982, −3.59598484286417298318871489887, −0.983455120709635509854134742943,
0.78421197113294179808088719083, 2.54833129716731088388915949514, 3.77207014312411371127763619319, 5.22514927195149906256989867533, 5.61671873907750438714187896428, 7.13467324613272497434833515342, 7.80035966628652641990590183770, 9.168150551779661874470487163260, 9.831219254563904522247018656449, 10.72706637634793288554328827306
