| L(s) = 1 | + (−0.104 + 0.994i)2-s + (0.882 + 1.49i)3-s + (−0.978 − 0.207i)4-s + (1.35 − 3.04i)5-s + (−1.57 + 0.721i)6-s + (1.07 − 2.41i)7-s + (0.309 − 0.951i)8-s + (−1.44 + 2.62i)9-s + (2.89 + 1.66i)10-s + (0.617 − 3.25i)11-s + (−0.552 − 1.64i)12-s + (1.75 − 2.41i)13-s + (2.29 + 1.32i)14-s + (5.74 − 0.666i)15-s + (0.913 + 0.406i)16-s + (−0.284 − 2.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (0.509 + 0.860i)3-s + (−0.489 − 0.103i)4-s + (0.607 − 1.36i)5-s + (−0.642 + 0.294i)6-s + (0.406 − 0.913i)7-s + (0.109 − 0.336i)8-s + (−0.481 + 0.876i)9-s + (0.914 + 0.527i)10-s + (0.186 − 0.982i)11-s + (−0.159 − 0.473i)12-s + (0.486 − 0.669i)13-s + (0.612 + 0.353i)14-s + (1.48 − 0.172i)15-s + (0.228 + 0.101i)16-s + (−0.0690 − 0.657i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69447 + 0.273134i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69447 + 0.273134i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.882 - 1.49i)T \) |
| 7 | \( 1 + (-1.07 + 2.41i)T \) |
| 11 | \( 1 + (-0.617 + 3.25i)T \) |
| good | 5 | \( 1 + (-1.35 + 3.04i)T + (-3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-1.75 + 2.41i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.284 + 2.70i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.31 - 6.16i)T + (-17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (0.928 - 0.536i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.535 + 1.64i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.61 + 2.05i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (6.72 - 7.47i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (3.40 - 10.4i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 0.998iT - 43T^{2} \) |
| 47 | \( 1 + (-1.94 - 9.14i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-2.79 - 6.28i)T + (-35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (1.23 - 5.83i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-4.16 + 9.34i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-1.86 + 3.22i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.722 + 0.994i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.24 + 5.85i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-17.1 - 1.79i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-9.70 + 7.05i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (11.7 - 6.78i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.77 + 2.01i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.79554236533691736959257844264, −9.930896393177555677353393693297, −9.259239305578384454813435738770, −8.187760027000126841507728126400, −8.007222086488119670687302392623, −6.19749128984127121010741164057, −5.28997406131368856322434865580, −4.51705574056637894639010641903, −3.42143510822943202438199472168, −1.18817886071671856652684089685,
1.91699215096781989766321264094, 2.45910286888733917012803376323, 3.70935936110517433547941272837, 5.38459234244793707889392344185, 6.62345584513730487048727913571, 7.15493207514092852648688719747, 8.535879294732491874735392738109, 9.162915989103239583863621316069, 10.17356163532185368214156572114, 11.09629985068903530710870721041
