| L(s) = 1 | + (−0.599 − 1.28i)2-s + (0.198 + 1.31i)3-s + (−1.28 + 1.53i)4-s + (3.14 + 1.23i)5-s + (1.56 − 1.04i)6-s + (−0.568 − 2.58i)7-s + (2.73 + 0.720i)8-s + (1.17 − 0.361i)9-s + (−0.304 − 4.76i)10-s + (−1.85 + 6.02i)11-s + (−2.27 − 1.38i)12-s + (−1.89 + 0.431i)13-s + (−2.96 + 2.27i)14-s + (−1.00 + 4.38i)15-s + (−0.717 − 3.93i)16-s + (0.911 − 0.621i)17-s + ⋯ |
| L(s) = 1 | + (−0.423 − 0.905i)2-s + (0.114 + 0.760i)3-s + (−0.640 + 0.767i)4-s + (1.40 + 0.552i)5-s + (0.639 − 0.425i)6-s + (−0.214 − 0.976i)7-s + (0.967 + 0.254i)8-s + (0.391 − 0.120i)9-s + (−0.0963 − 1.50i)10-s + (−0.560 + 1.81i)11-s + (−0.657 − 0.398i)12-s + (−0.524 + 0.119i)13-s + (−0.793 + 0.608i)14-s + (−0.258 + 1.13i)15-s + (−0.179 − 0.983i)16-s + (0.221 − 0.150i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33172 + 0.146027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33172 + 0.146027i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.599 + 1.28i)T \) |
| 7 | \( 1 + (0.568 + 2.58i)T \) |
| good | 3 | \( 1 + (-0.198 - 1.31i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-3.14 - 1.23i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (1.85 - 6.02i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (1.89 - 0.431i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.911 + 0.621i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.56 + 0.900i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.57 - 1.75i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-1.18 + 2.45i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (2.65 - 4.60i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-11.4 + 0.859i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-0.342 - 0.429i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (7.19 + 5.73i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-8.23 + 7.64i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (3.47 + 0.260i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (5.41 - 2.12i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (5.46 - 0.409i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (10.8 + 6.25i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.4 + 5.03i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-4.67 - 4.33i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (3.05 + 5.28i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.53 + 1.26i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (12.0 - 3.70i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 8.69T + 97T^{2} \) |
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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
−10.83526611786984020670443786209, −10.23264747732775149664376763348, −9.750050405879509722882698796087, −9.305059120576688878206197845088, −7.52802598944578653393918708079, −6.91254079719963447845085909958, −5.10911176546659425283694777847, −4.28452793342257445767685555330, −2.93707205414501805229586214901, −1.73432730361634576487135848250,
1.16868930211398344020345547001, 2.61621849532736687553980122859, 4.94595855904943066460880457540, 5.87760487430386545487021802906, 6.24972091106344898399518059050, 7.65914962480643582062746979470, 8.458819771532517053562729448342, 9.278191338603571802884046097561, 9.980728344595737460147292016098, 11.10593955176646878653313009728
