| L(s) = 1 | + (−311. − 540. i)3-s + (−5.29e3 + 9.17e3i)5-s + (−2.36e4 + 3.76e4i)7-s + (−1.05e5 + 1.83e5i)9-s + (3.40e5 + 5.89e5i)11-s + 8.21e5·13-s + 6.60e6·15-s + (2.07e6 + 3.58e6i)17-s + (−5.48e5 + 9.49e5i)19-s + (2.77e7 + 1.03e6i)21-s + (−5.52e6 + 9.57e6i)23-s + (−3.16e7 − 5.48e7i)25-s + 2.15e7·27-s − 1.24e7·29-s + (1.22e8 + 2.12e8i)31-s + ⋯ |
| L(s) = 1 | + (−0.740 − 1.28i)3-s + (−0.757 + 1.31i)5-s + (−0.532 + 0.846i)7-s + (−0.597 + 1.03i)9-s + (0.637 + 1.10i)11-s + 0.613·13-s + 2.24·15-s + (0.353 + 0.612i)17-s + (−0.0508 + 0.0880i)19-s + (1.48 + 0.0555i)21-s + (−0.179 + 0.310i)23-s + (−0.648 − 1.12i)25-s + 0.289·27-s − 0.112·29-s + (0.769 + 1.33i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.8151036882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8151036882\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.36e4 - 3.76e4i)T \) |
| good | 3 | \( 1 + (311. + 540. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + (5.29e3 - 9.17e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-3.40e5 - 5.89e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 8.21e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-2.07e6 - 3.58e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (5.48e5 - 9.49e5i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (5.52e6 - 9.57e6i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.24e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.22e8 - 2.12e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-7.61e6 + 1.31e7i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 3.80e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 9.14e6T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-7.65e8 + 1.32e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-2.54e9 - 4.41e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-3.31e9 - 5.74e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (6.13e8 - 1.06e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-4.16e9 - 7.21e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 2.83e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-7.48e9 - 1.29e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (1.08e10 - 1.87e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 4.11e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (2.64e10 - 4.57e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 5.72e10T + 7.15e21T^{2} \) |
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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
−12.05186206721746405225562961252, −11.17077364237914330595245472496, −10.02367111510751194552337686224, −8.421686091168222558419362359992, −7.15998782231926123616614228937, −6.69030648407471183728524198151, −5.70637919483651891505434710029, −3.79710409865556424128834601349, −2.50179075615433273108557907378, −1.29494799277843537013013948521,
0.29564892646728644703314345268, 0.858967027956853168304779222900, 3.56203548660588652389670847885, 4.17681107999709888089437966562, 5.17556591007005610578805060988, 6.33098192394187503095969892224, 8.021121218523238169758062892694, 9.073101690562622746390640834169, 9.962643619311957027824742457905, 11.13211883923225144879587397634
