| L(s) = 1 | + (0.913 + 0.406i)2-s + (1.57 + 0.726i)3-s + (0.669 + 0.743i)4-s + (0.913 − 0.406i)5-s + (1.14 + 1.30i)6-s + (−4.74 + 1.00i)7-s + (0.309 + 0.951i)8-s + (1.94 + 2.28i)9-s + 0.999·10-s + (0.272 + 3.30i)11-s + (0.512 + 1.65i)12-s + (0.108 + 1.03i)13-s + (−4.74 − 1.00i)14-s + (1.73 + 0.0239i)15-s + (−0.104 + 0.994i)16-s + (5.36 + 3.89i)17-s + ⋯ |
| L(s) = 1 | + (0.645 + 0.287i)2-s + (0.907 + 0.419i)3-s + (0.334 + 0.371i)4-s + (0.408 − 0.181i)5-s + (0.465 + 0.531i)6-s + (−1.79 + 0.381i)7-s + (0.109 + 0.336i)8-s + (0.648 + 0.761i)9-s + 0.316·10-s + (0.0821 + 0.996i)11-s + (0.147 + 0.477i)12-s + (0.0300 + 0.285i)13-s + (−1.26 − 0.269i)14-s + (0.447 + 0.00619i)15-s + (−0.0261 + 0.248i)16-s + (1.30 + 0.944i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89610 + 2.09712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89610 + 2.09712i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 3 | \( 1 + (-1.57 - 0.726i)T \) |
| 5 | \( 1 + (-0.913 + 0.406i)T \) |
| 11 | \( 1 + (-0.272 - 3.30i)T \) |
| good | 7 | \( 1 + (4.74 - 1.00i)T + (6.39 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-0.108 - 1.03i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-5.36 - 3.89i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.19 + 3.67i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.73 + 3.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (6.60 - 1.40i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.789 - 7.50i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-1.14 + 3.51i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-10.9 - 2.32i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (4.30 + 7.45i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.86 + 3.18i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.579 + 0.420i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (8.19 + 9.09i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.348 - 3.32i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-2.20 + 3.81i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.537 + 0.390i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.15 + 6.64i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (2.67 + 1.19i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.785 + 7.47i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 - 9.51T + 89T^{2} \) |
| 97 | \( 1 + (-6.30 - 2.80i)T + (64.9 + 72.0i)T^{2} \) |
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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
−10.03844413524460670336542796851, −9.318494461842268060702159380699, −8.775107506121594745518270428252, −7.52610147566543837370750537911, −6.80139349903870954354780045604, −5.92530521303647969889643793906, −4.90575053771905743129535439177, −3.82548066413822653006304426316, −3.08947235554075746818040566353, −2.05876934937745311983403111415,
0.969285488778915751873873452520, 2.65643118636754747396458862144, 3.27607922618149429968344915011, 3.96599912526925769856246616288, 5.75970988829280702374720724695, 6.16547448216572019732392316218, 7.22050841379402982198163263822, 7.902874710530515174975304283675, 9.367455742327465413900530786611, 9.578652853370363100130886363394
