| 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
−10.73377234650123686671805307169, −9.817274301881477499673985293704, −9.524279282382425575744255372221, −8.358379037679780923475270210583, −7.79373459170892667681815916397, −6.47848425083809156698378331806, −4.95795848604755847817917948024, −4.25132758956669525278729642060, −3.06565294718463831985627721253, −1.71340514822483369395940450672,
0.830601328343011107980689550292, 2.40421374748610930045555960511, 3.62449828381307761715969483114, 5.40240927049356005107236454934, 6.32467193903955482577515699724, 6.97294180477614429162629211154, 8.241835773127579717499412294414, 8.402492705551256165911323087855, 9.500260766710267498717068654035, 10.58965070528971289289163450905
