| L(s) = 1 | + (0.273 − 1.89i)2-s + (−1.34 − 2.94i)3-s + (−1.61 − 0.474i)4-s + (0.654 + 0.755i)5-s + (−5.95 + 1.74i)6-s + (3.10 + 1.99i)7-s + (0.253 − 0.554i)8-s + (−4.88 + 5.64i)9-s + (1.61 − 1.03i)10-s + (−0.00937 − 0.0652i)11-s + (0.774 + 5.38i)12-s + (0.395 − 0.253i)13-s + (4.63 − 5.34i)14-s + (1.34 − 2.94i)15-s + (−3.81 − 2.45i)16-s + (−4.79 + 1.40i)17-s + ⋯ |
| L(s) = 1 | + (0.193 − 1.34i)2-s + (−0.775 − 1.69i)3-s + (−0.807 − 0.237i)4-s + (0.292 + 0.337i)5-s + (−2.43 + 0.713i)6-s + (1.17 + 0.753i)7-s + (0.0894 − 0.195i)8-s + (−1.62 + 1.88i)9-s + (0.510 − 0.328i)10-s + (−0.00282 − 0.0196i)11-s + (0.223 + 1.55i)12-s + (0.109 − 0.0704i)13-s + (1.23 − 1.42i)14-s + (0.346 − 0.759i)15-s + (−0.953 − 0.612i)16-s + (−1.16 + 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.151524 - 1.01766i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151524 - 1.01766i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-0.809 + 4.72i)T \) |
| good | 2 | \( 1 + (-0.273 + 1.89i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (1.34 + 2.94i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (-3.10 - 1.99i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.00937 + 0.0652i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.395 + 0.253i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.79 - 1.40i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-4.37 - 1.28i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.472 + 0.138i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (3.09 - 6.77i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.29 + 2.64i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.51 - 1.74i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.64 - 3.59i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 0.621T + 47T^{2} \) |
| 53 | \( 1 + (3.03 + 1.94i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-4.46 + 2.86i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (4.24 - 9.29i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.198 + 1.37i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.52 + 10.6i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (14.2 + 4.18i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (2.48 - 1.59i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-3.52 + 4.06i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.97 - 6.51i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.09 - 2.42i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−12.76782997103053635212915278037, −12.01605427394863724454462205439, −11.30250724472856533266328788937, −10.65266525713463986397411037528, −8.788060943729270277239210426325, −7.51039876087496054287297106700, −6.31891942469458079764983764833, −5.00255548357574899664463337585, −2.53265768285240510002412189925, −1.50696570994404408520774412602,
4.22392913414417051622698697540, 4.98166352983004710614024401398, 5.84498273372566473985575711955, 7.32147585794993573296404010766, 8.696678508225142763639571327567, 9.700899859454505568566559211652, 11.04460266864581180418890206334, 11.45729740732817863593725910554, 13.56014716462226695242757811568, 14.40121508316163115073082532325
