L(s) = 1 | + (0.451 + 0.146i)3-s + (2.19 − 0.420i)5-s − 3.03i·7-s + (−2.24 − 1.63i)9-s + (1.61 − 1.17i)11-s + (0.838 − 1.15i)13-s + (1.05 + 0.132i)15-s + (−1.76 + 0.574i)17-s + (0.279 + 0.859i)19-s + (0.445 − 1.37i)21-s + (1.95 + 2.69i)23-s + (4.64 − 1.84i)25-s + (−1.61 − 2.21i)27-s + (−1.22 + 3.76i)29-s + (1.99 + 6.12i)31-s + ⋯ |
L(s) = 1 | + (0.260 + 0.0847i)3-s + (0.982 − 0.187i)5-s − 1.14i·7-s + (−0.748 − 0.543i)9-s + (0.487 − 0.354i)11-s + (0.232 − 0.320i)13-s + (0.272 + 0.0342i)15-s + (−0.429 + 0.139i)17-s + (0.0640 + 0.197i)19-s + (0.0972 − 0.299i)21-s + (0.408 + 0.561i)23-s + (0.929 − 0.369i)25-s + (−0.310 − 0.426i)27-s + (−0.227 + 0.699i)29-s + (0.357 + 1.10i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.54992 - 0.612760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.54992 - 0.612760i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.19 + 0.420i)T \) |
good | 3 | \( 1 + (-0.451 - 0.146i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 3.03iT - 7T^{2} \) |
| 11 | \( 1 + (-1.61 + 1.17i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.838 + 1.15i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.76 - 0.574i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.279 - 0.859i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.95 - 2.69i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.22 - 3.76i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.99 - 6.12i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.24 + 3.09i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.48 - 1.07i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 3.59iT - 43T^{2} \) |
| 47 | \( 1 + (-4.56 - 1.48i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-9.03 - 2.93i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (8.61 + 6.25i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (11.5 - 8.39i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (10.1 - 3.30i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (3.85 - 11.8i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.157 - 0.216i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.64 - 8.15i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-12.0 + 3.89i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (3.85 - 2.80i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (9.47 + 3.07i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.94357546353758288530681325061, −10.31840895900490984491095608021, −9.227641124370688552940319895275, −8.676360225945228971047304117332, −7.36418289243737067955436561634, −6.37322799403794260804789829484, −5.47159040537015728590700168524, −4.09360755491264603908229055846, −2.96403140240530438553489442872, −1.20021054434321840386314405228,
2.02228619510723260234874260690, 2.83777626156633201344940344717, 4.64774103325362431985529921718, 5.77252526444773650668764909694, 6.42287224951169299077206951636, 7.76314501444866887230759132681, 8.960099087257332897132118213391, 9.248107812376861081817028339414, 10.46810426920174511142344877039, 11.41376948171853765311030070112