| L(s) = 1 | + (−0.234 − 0.0854i)2-s + (−0.399 − 2.26i)3-s + (−1.48 − 1.24i)4-s + (−0.0998 + 0.566i)6-s + (1.98 − 3.42i)7-s + (0.491 + 0.851i)8-s + (−2.15 + 0.785i)9-s + (−1.56 − 2.71i)11-s + (−2.22 + 3.86i)12-s + (0.461 − 2.61i)13-s + (−0.757 + 0.635i)14-s + (0.630 + 3.57i)16-s + (2.12 + 0.771i)17-s + 0.573·18-s + (−3.76 + 2.19i)19-s + ⋯ |
| L(s) = 1 | + (−0.165 − 0.0604i)2-s + (−0.230 − 1.30i)3-s + (−0.742 − 0.622i)4-s + (−0.0407 + 0.231i)6-s + (0.748 − 1.29i)7-s + (0.173 + 0.301i)8-s + (−0.719 + 0.261i)9-s + (−0.472 − 0.818i)11-s + (−0.643 + 1.11i)12-s + (0.128 − 0.726i)13-s + (−0.202 + 0.169i)14-s + (0.157 + 0.893i)16-s + (0.514 + 0.187i)17-s + 0.135·18-s + (−0.864 + 0.502i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0166596 + 0.926715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0166596 + 0.926715i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (3.76 - 2.19i)T \) |
| good | 2 | \( 1 + (0.234 + 0.0854i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (0.399 + 2.26i)T + (-2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (-1.98 + 3.42i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.56 + 2.71i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.461 + 2.61i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.12 - 0.771i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-5.81 - 4.87i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.28 - 0.466i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.447 - 0.774i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.62T + 37T^{2} \) |
| 41 | \( 1 + (-1.08 - 6.14i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.35 + 1.13i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.165 + 0.0603i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (5.19 + 4.35i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-7.34 - 2.67i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.796 + 0.668i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-14.2 + 5.20i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-7.99 + 6.70i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.726 + 4.11i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.94 - 11.0i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.62 + 11.4i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.257 + 1.46i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (14.4 + 5.26i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.75065387649719988094757796540, −9.795523976299238767428059328728, −8.402571373519563425216608753637, −7.910290243458088467147480955246, −6.97643322470069312471309826568, −5.86534245804165982361626392742, −4.99135609254063845860635259715, −3.61325467066321765415079527525, −1.59710750026121920572451854721, −0.67428124087338899081659004718,
2.45169912899588953393931673197, 3.91929370336331499762016854345, 4.84113757833838007062242093794, 5.30327558389759296234680594172, 6.94668338985421540927493282719, 8.209375296886093130959933183228, 8.955028751021140285554051208711, 9.472150031462229559285824036548, 10.49157020385561234397361314832, 11.33126580148993042951694451671
