| L(s) = 1 | + (1.06 − 1.18i)2-s + (0.104 − 0.994i)3-s + (−0.0561 − 0.533i)4-s + (−1.11 + 3.44i)5-s + (−1.06 − 1.18i)6-s + (0.311 + 2.95i)7-s + (1.88 + 1.36i)8-s + (−0.978 − 0.207i)9-s + (2.88 + 4.99i)10-s + (0.0865 − 3.31i)11-s − 0.536·12-s + (2.99 + 2.00i)13-s + (3.83 + 2.78i)14-s + (3.31 + 1.47i)15-s + (4.68 − 0.994i)16-s + (−2.23 − 2.48i)17-s + ⋯ |
| L(s) = 1 | + (0.753 − 0.836i)2-s + (0.0603 − 0.574i)3-s + (−0.0280 − 0.266i)4-s + (−0.500 + 1.54i)5-s + (−0.435 − 0.483i)6-s + (0.117 + 1.11i)7-s + (0.666 + 0.484i)8-s + (−0.326 − 0.0693i)9-s + (0.912 + 1.58i)10-s + (0.0261 − 0.999i)11-s − 0.154·12-s + (0.831 + 0.555i)13-s + (1.02 + 0.744i)14-s + (0.854 + 0.380i)15-s + (1.17 − 0.248i)16-s + (−0.542 − 0.602i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.00205 + 0.0268830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00205 + 0.0268830i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 11 | \( 1 + (-0.0865 + 3.31i)T \) |
| 13 | \( 1 + (-2.99 - 2.00i)T \) |
| good | 2 | \( 1 + (-1.06 + 1.18i)T + (-0.209 - 1.98i)T^{2} \) |
| 5 | \( 1 + (1.11 - 3.44i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-0.311 - 2.95i)T + (-6.84 + 1.45i)T^{2} \) |
| 17 | \( 1 + (2.23 + 2.48i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.0179 + 0.00797i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-4.09 - 7.08i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.61 + 1.60i)T + (19.4 + 21.5i)T^{2} \) |
| 31 | \( 1 + (1.67 + 5.16i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.26 - 2.34i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (-0.969 + 9.22i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (2.57 - 4.45i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.83 + 2.06i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.22 + 3.75i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (1.08 + 10.2i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (4.88 + 5.42i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-5.21 - 9.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.10 - 3.44i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-8.36 + 6.08i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.35 - 7.24i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.235 + 0.726i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (6.15 + 10.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.97 - 2.11i)T + (88.6 + 39.4i)T^{2} \) |
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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
−11.36346684305453394729017351500, −10.98829935314369715865430785322, −9.435578035266484494969666285192, −8.317523287452560729495946727771, −7.41863316469656827287037228897, −6.39363031982956567506010604999, −5.42638145760966241035234797512, −3.76296114576727488524863800330, −3.03860704399403235320014522809, −2.08507775587720232748439813631,
1.14121261432293229572878871157, 3.83556034402641701686050203272, 4.51390689849757853841113554479, 5.06716758699320466107685749455, 6.32802551105378845962115182945, 7.42172329384200949505825287094, 8.281341197883311900252144900602, 9.205264066409304201992162691634, 10.35033987939207279979414362535, 10.99840523188521447982680923340
