| L(s) = 1 | + (−0.530 − 1.31i)2-s + (−0.112 − 0.346i)3-s + (−1.43 + 1.39i)4-s + (1.22 + 1.86i)5-s + (−0.394 + 0.331i)6-s + 4.30i·7-s + (2.58 + 1.14i)8-s + (2.31 − 1.68i)9-s + (1.79 − 2.60i)10-s + (−1.67 + 2.30i)11-s + (0.643 + 0.341i)12-s + (1.64 − 1.19i)13-s + (5.63 − 2.28i)14-s + (0.508 − 0.635i)15-s + (0.129 − 3.99i)16-s + (−6.70 − 2.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.375 − 0.926i)2-s + (−0.0649 − 0.199i)3-s + (−0.718 + 0.695i)4-s + (0.549 + 0.835i)5-s + (−0.160 + 0.135i)6-s + 1.62i·7-s + (0.914 + 0.404i)8-s + (0.773 − 0.561i)9-s + (0.568 − 0.822i)10-s + (−0.504 + 0.694i)11-s + (0.185 + 0.0984i)12-s + (0.455 − 0.330i)13-s + (1.50 − 0.609i)14-s + (0.131 − 0.164i)15-s + (0.0322 − 0.999i)16-s + (−1.62 − 0.528i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00086 - 0.00354033i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00086 - 0.00354033i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.530 + 1.31i)T \) |
| 5 | \( 1 + (-1.22 - 1.86i)T \) |
| good | 3 | \( 1 + (0.112 + 0.346i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 4.30iT - 7T^{2} \) |
| 11 | \( 1 + (1.67 - 2.30i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.64 + 1.19i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (6.70 + 2.17i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.54 - 1.80i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.945 + 1.30i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-7.60 + 2.47i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.685 - 2.10i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.672 - 0.488i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.65 + 5.56i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 5.08T + 43T^{2} \) |
| 47 | \( 1 + (0.833 - 0.270i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.578 - 1.78i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (8.01 + 11.0i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (4.59 - 6.33i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.35 + 4.16i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.71 + 5.28i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.126 + 0.174i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.86 + 5.73i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.65 + 8.17i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.97 - 1.43i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.16 + 1.67i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.31646086087895576666179883752, −11.55179095091928421372810853959, −10.46858295636600934641925101902, −9.577878821511359458500449867995, −8.844360830765842784999361057957, −7.47243406414685856365104769380, −6.25324727236021665856040240611, −4.85840286446711267228202034640, −3.05504651201116406721501061131, −2.03267274145517838357772067033,
1.15574403704297973516210824098, 4.20874254134253404868771293879, 4.97578011353729648454310741109, 6.37410422301428608499336337069, 7.38167876264265724776600527546, 8.374493853780990740667052656625, 9.427016692402632730605042687483, 10.34595979407151733669877971157, 11.05382547687089167941960778182, 13.10843080522563002140192320443
