| L(s) = 1 | + (−0.402 + 0.258i)2-s + (0.142 − 0.989i)3-s + (−0.735 + 1.61i)4-s + (3.37 − 0.989i)5-s + (0.198 + 0.435i)6-s + (0.527 + 0.608i)7-s + (−0.256 − 1.78i)8-s + (−0.959 − 0.281i)9-s + (−1.10 + 1.27i)10-s + (−2.98 − 1.91i)11-s + (1.48 + 0.957i)12-s + (−4.32 + 4.99i)13-s + (−0.370 − 0.108i)14-s + (−0.499 − 3.47i)15-s + (−1.75 − 2.02i)16-s + (0.387 + 0.849i)17-s + ⋯ |
| L(s) = 1 | + (−0.284 + 0.183i)2-s + (0.0821 − 0.571i)3-s + (−0.367 + 0.805i)4-s + (1.50 − 0.442i)5-s + (0.0812 + 0.177i)6-s + (0.199 + 0.230i)7-s + (−0.0908 − 0.631i)8-s + (−0.319 − 0.0939i)9-s + (−0.348 + 0.402i)10-s + (−0.899 − 0.578i)11-s + (0.430 + 0.276i)12-s + (−1.20 + 1.38i)13-s + (−0.0989 − 0.0290i)14-s + (−0.129 − 0.897i)15-s + (−0.438 − 0.505i)16-s + (0.0940 + 0.206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.873202 + 0.0279402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.873202 + 0.0279402i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (4.71 + 0.863i)T \) |
| good | 2 | \( 1 + (0.402 - 0.258i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (-3.37 + 0.989i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.527 - 0.608i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (2.98 + 1.91i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (4.32 - 4.99i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.387 - 0.849i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.55 + 3.41i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.657 - 1.43i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.0804 + 0.559i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-7.84 - 2.30i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.65 + 1.07i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.975 - 6.78i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 5.69T + 47T^{2} \) |
| 53 | \( 1 + (4.57 + 5.27i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.663 - 0.766i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.74 - 12.1i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.09 + 1.34i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-11.1 + 7.16i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-2.78 + 6.09i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.940 - 1.08i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.68 - 0.493i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.667 + 4.64i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (0.762 - 0.223i)T + (81.6 - 52.4i)T^{2} \) |
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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
−14.41151605560174409731811854788, −13.52620014758144925739665169907, −12.83400578987676295386996054479, −11.69018416050556484736023511593, −9.838483789721755162171946523870, −9.019705076138058324199070184054, −7.83417673982626995601800418985, −6.43712617853384580873324239530, −4.93317686616211836923991637321, −2.40726528784854194911626272339,
2.39490186825623008111632867174, 5.04042543215895933950633249381, 5.87852267674029089422716125515, 7.86443462951314553868318407929, 9.744116546788113938058718479775, 9.918347415975715097438735236301, 10.82880856250491456254331467201, 12.73046024928202900521472096382, 13.95090513971500873735780887409, 14.57696106628472022154378435994
