| 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 | \( 1 + (0.762 + 0.223i)T + (81.6 + 52.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.57696106628472022154378435994, −13.95090513971500873735780887409, −12.73046024928202900521472096382, −10.82880856250491456254331467201, −9.918347415975715097438735236301, −9.744116546788113938058718479775, −7.86443462951314553868318407929, −5.87852267674029089422716125515, −5.04042543215895933950633249381, −2.39490186825623008111632867174,
2.40726528784854194911626272339, 4.93317686616211836923991637321, 6.43712617853384580873324239530, 7.83417673982626995601800418985, 9.019705076138058324199070184054, 9.838483789721755162171946523870, 11.69018416050556484736023511593, 12.83400578987676295386996054479, 13.52620014758144925739665169907, 14.41151605560174409731811854788
