| L(s) = 1 | + (−1.02 − 0.973i)2-s + (−0.540 + 0.841i)3-s + (0.105 + 1.99i)4-s + (2.63 + 1.20i)5-s + (1.37 − 0.336i)6-s + (3.10 + 0.910i)7-s + (1.83 − 2.15i)8-s + (−0.415 − 0.909i)9-s + (−1.53 − 3.80i)10-s + (1.24 + 1.08i)11-s + (−1.73 − 0.991i)12-s + (0.884 + 3.01i)13-s + (−2.29 − 3.95i)14-s + (−2.44 + 1.56i)15-s + (−3.97 + 0.420i)16-s + (−0.685 − 4.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.725 − 0.688i)2-s + (−0.312 + 0.485i)3-s + (0.0525 + 0.998i)4-s + (1.18 + 0.539i)5-s + (0.560 − 0.137i)6-s + (1.17 + 0.344i)7-s + (0.649 − 0.760i)8-s + (−0.138 − 0.303i)9-s + (−0.485 − 1.20i)10-s + (0.376 + 0.326i)11-s + (−0.501 − 0.286i)12-s + (0.245 + 0.835i)13-s + (−0.613 − 1.05i)14-s + (−0.630 + 0.405i)15-s + (−0.994 + 0.105i)16-s + (−0.166 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22896 + 0.311373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22896 + 0.311373i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.02 + 0.973i)T \) |
| 3 | \( 1 + (0.540 - 0.841i)T \) |
| 23 | \( 1 + (4.70 - 0.912i)T \) |
| good | 5 | \( 1 + (-2.63 - 1.20i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-3.10 - 0.910i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.24 - 1.08i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.884 - 3.01i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.685 + 4.76i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.389 - 0.0560i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-4.10 + 0.589i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (4.60 - 2.95i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.39 + 1.54i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.52 + 3.33i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (5.82 - 9.06i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 8.73T + 47T^{2} \) |
| 53 | \( 1 + (-3.74 + 12.7i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.25 - 4.26i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.95 - 6.15i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (1.33 - 1.15i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (8.80 + 10.1i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.408 + 2.84i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (1.26 - 0.370i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (7.35 - 3.35i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (0.761 + 0.489i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (4.43 - 9.70i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.78777552888125249505675197516, −9.955066566750479884013773623967, −9.339642779774175072497269858886, −8.543124174652130549007509641453, −7.34001449132905182621197685712, −6.37095294368149980250199090544, −5.15866575692299119304471272504, −4.11035832103022084146923507574, −2.56526813362670657060328087306, −1.61975891664483333095028557937,
1.11903780627735096141371121153, 2.00950772483988480876605594085, 4.43861994598532541515186646265, 5.64934052063206525186116982349, 5.95203485342915809939556798332, 7.18473736121213398884032578112, 8.204269332638147819965713271199, 8.651501319148057114153262677875, 9.832244627544609533725314049016, 10.56323297469228870857324687804
