| L(s) = 1 | + (−0.743 + 0.669i)2-s + (0.746 − 1.56i)3-s + (0.104 − 0.994i)4-s + 0.690i·5-s + (0.491 + 1.66i)6-s + (0.487 + 0.354i)7-s + (0.587 + 0.809i)8-s + (−1.88 − 2.33i)9-s + (−0.461 − 0.512i)10-s + (0.468 − 4.45i)11-s + (−1.47 − 0.905i)12-s + (1.31 − 0.427i)13-s + (−0.599 + 0.0629i)14-s + (1.07 + 0.515i)15-s + (−0.978 − 0.207i)16-s + (−0.697 − 0.310i)17-s + ⋯ |
| L(s) = 1 | + (−0.525 + 0.473i)2-s + (0.430 − 0.902i)3-s + (0.0522 − 0.497i)4-s + 0.308i·5-s + (0.200 + 0.678i)6-s + (0.184 + 0.133i)7-s + (0.207 + 0.286i)8-s + (−0.628 − 0.777i)9-s + (−0.146 − 0.162i)10-s + (0.141 − 1.34i)11-s + (−0.426 − 0.261i)12-s + (0.364 − 0.118i)13-s + (−0.160 + 0.0168i)14-s + (0.278 + 0.132i)15-s + (−0.244 − 0.0519i)16-s + (−0.169 − 0.0753i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01436 - 0.667779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01436 - 0.667779i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.743 - 0.669i)T \) |
| 3 | \( 1 + (-0.746 + 1.56i)T \) |
| 31 | \( 1 + (-3.33 + 4.45i)T \) |
| good | 5 | \( 1 - 0.690iT - 5T^{2} \) |
| 7 | \( 1 + (-0.487 - 0.354i)T + (2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (-0.468 + 4.45i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.31 + 0.427i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.697 + 0.310i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-2.78 + 0.591i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (6.82 + 3.04i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (-3.22 - 3.57i)T + (-3.03 + 28.8i)T^{2} \) |
| 37 | \( 1 + (-6.17 + 3.56i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.93 + 1.92i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (7.40 + 2.40i)T + (34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-1.89 + 8.91i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-0.410 - 3.90i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (1.23 - 5.82i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (0.940 + 0.542i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 - 5.05T + 67T^{2} \) |
| 71 | \( 1 + (-11.3 + 1.19i)T + (69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-5.22 - 11.7i)T + (-48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-4.56 - 6.27i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (3.23 - 0.687i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-11.5 - 8.35i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (9.86 - 4.39i)T + (64.9 - 72.0i)T^{2} \) |
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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
−10.58965070528971289289163450905, −9.500260766710267498717068654035, −8.402492705551256165911323087855, −8.241835773127579717499412294414, −6.97294180477614429162629211154, −6.32467193903955482577515699724, −5.40240927049356005107236454934, −3.62449828381307761715969483114, −2.40421374748610930045555960511, −0.830601328343011107980689550292,
1.71340514822483369395940450672, 3.06565294718463831985627721253, 4.25132758956669525278729642060, 4.95795848604755847817917948024, 6.47848425083809156698378331806, 7.79373459170892667681815916397, 8.358379037679780923475270210583, 9.524279282382425575744255372221, 9.817274301881477499673985293704, 10.73377234650123686671805307169
