| L(s) = 1 | + (1.33 + 0.475i)2-s + (0.654 + 0.755i)3-s + (1.54 + 1.26i)4-s + (−0.462 − 3.21i)5-s + (0.513 + 1.31i)6-s + (1.43 − 3.13i)7-s + (1.46 + 2.42i)8-s + (−0.142 + 0.989i)9-s + (0.913 − 4.50i)10-s + (0.0298 − 0.101i)11-s + (0.0568 + 1.99i)12-s + (−0.610 + 0.278i)13-s + (3.39 − 3.49i)14-s + (2.12 − 2.45i)15-s + (0.793 + 3.92i)16-s + (0.598 − 0.930i)17-s + ⋯ |
| L(s) = 1 | + (0.941 + 0.336i)2-s + (0.378 + 0.436i)3-s + (0.774 + 0.633i)4-s + (−0.206 − 1.43i)5-s + (0.209 + 0.538i)6-s + (0.540 − 1.18i)7-s + (0.516 + 0.856i)8-s + (−0.0474 + 0.329i)9-s + (0.288 − 1.42i)10-s + (0.00898 − 0.0306i)11-s + (0.0164 + 0.577i)12-s + (−0.169 + 0.0773i)13-s + (0.907 − 0.933i)14-s + (0.549 − 0.634i)15-s + (0.198 + 0.980i)16-s + (0.145 − 0.225i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.89890 + 0.0564619i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.89890 + 0.0564619i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.33 - 0.475i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (2.35 + 4.17i)T \) |
| good | 5 | \( 1 + (0.462 + 3.21i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.43 + 3.13i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.0298 + 0.101i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.610 - 0.278i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.598 + 0.930i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-2.54 - 3.96i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (2.63 - 4.10i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.0721 + 0.0624i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.960 - 6.67i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.346 - 2.41i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.63 + 1.41i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 8.18iT - 47T^{2} \) |
| 53 | \( 1 + (2.84 - 6.22i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (6.33 + 13.8i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (3.23 - 3.73i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-3.45 - 11.7i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (2.59 + 8.82i)T + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (1.31 - 0.842i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (5.50 + 12.0i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (15.2 + 2.19i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-11.9 + 10.3i)T + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-8.96 + 1.28i)T + (93.0 - 27.3i)T^{2} \) |
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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
−10.90285440397801831987230358011, −9.973594616450815541251382188462, −8.797939566419672307810855450252, −7.984632895533162446745116842824, −7.36700828525731252200418802508, −5.94409975541080716197515861325, −4.77729803777921597774204721754, −4.43412554305774757205935395868, −3.34060322426904079130687333577, −1.48990423401963145226535218576,
2.03659711553332369764040248366, 2.80197345995691995218733090317, 3.79534989379072103845762409783, 5.29115503881306798912130175226, 6.10669156609967771046804969388, 7.09917780438835393972898127656, 7.77546082622422780348147654715, 9.121649776214680666327687211391, 10.11726898806841804369542990286, 11.14665260497563839749721689701
