| L(s) = 1 | + (−0.606 − 0.270i)2-s + (1.10 + 1.33i)3-s + (−1.04 − 1.15i)4-s + (1.02 − 0.454i)5-s + (−0.308 − 1.10i)6-s + (3.81 − 0.811i)7-s + (0.730 + 2.24i)8-s + (−0.568 + 2.94i)9-s − 0.741·10-s + (−2.09 + 2.57i)11-s + (0.397 − 2.67i)12-s + (−0.279 − 2.66i)13-s + (−2.53 − 0.539i)14-s + (1.73 + 0.861i)15-s + (−0.161 + 1.53i)16-s + (−4.48 − 3.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.429 − 0.191i)2-s + (0.636 + 0.771i)3-s + (−0.521 − 0.579i)4-s + (0.456 − 0.203i)5-s + (−0.125 − 0.452i)6-s + (1.44 − 0.306i)7-s + (0.258 + 0.794i)8-s + (−0.189 + 0.981i)9-s − 0.234·10-s + (−0.631 + 0.775i)11-s + (0.114 − 0.770i)12-s + (−0.0776 − 0.738i)13-s + (−0.678 − 0.144i)14-s + (0.447 + 0.222i)15-s + (−0.0404 + 0.384i)16-s + (−1.08 − 0.790i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.989026 + 0.00452261i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.989026 + 0.00452261i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.10 - 1.33i)T \) |
| 11 | \( 1 + (2.09 - 2.57i)T \) |
| good | 2 | \( 1 + (0.606 + 0.270i)T + (1.33 + 1.48i)T^{2} \) |
| 5 | \( 1 + (-1.02 + 0.454i)T + (3.34 - 3.71i)T^{2} \) |
| 7 | \( 1 + (-3.81 + 0.811i)T + (6.39 - 2.84i)T^{2} \) |
| 13 | \( 1 + (0.279 + 2.66i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (4.48 + 3.26i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.58 + 4.88i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.05 - 1.83i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.05 - 0.225i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.0449 - 0.427i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (2.69 - 8.27i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.85 - 0.606i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-1.11 - 1.93i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.58 + 9.53i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (4.56 - 3.31i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.58 + 1.76i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.426 + 4.06i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-4.87 + 8.44i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.11 + 3.71i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.60 - 11.1i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.12 - 1.83i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (0.0503 - 0.479i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + (-7.32 - 3.26i)T + (64.9 + 72.0i)T^{2} \) |
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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
−13.89230862253041592813924103767, −13.31096014528052694144703969745, −11.28948150926369198592247154366, −10.54008808553636230682830981737, −9.569608519041224589345820713829, −8.657274510287868079531999776354, −7.63063279267574852682238503467, −5.19507308577847177458293116040, −4.62585654673507576368523935942, −2.12755173684195276252197362051,
2.10736496354344219624781801785, 4.14312953230187697805103583179, 6.05178045032888392529724040206, 7.58011989397934821464363257397, 8.358233319291793301185429639500, 9.065795320490033464341255836990, 10.65960587532405390345054805672, 11.97134595202064047869291120279, 12.98903686804784162881212265153, 13.97737746023028621357320882561
