| L(s) = 1 | + (−0.841 − 0.540i)3-s + (−0.182 + 0.399i)5-s + (−0.331 − 0.0973i)7-s + (0.415 + 0.909i)9-s + (−3.78 + 4.37i)11-s + (−1.15 + 0.338i)13-s + (0.369 − 0.237i)15-s + (−0.949 − 6.60i)17-s + (−0.647 + 4.50i)19-s + (0.226 + 0.261i)21-s + (−4.20 + 2.30i)23-s + (3.14 + 3.63i)25-s + (0.142 − 0.989i)27-s + (1.52 + 10.5i)29-s + (−0.700 + 0.450i)31-s + ⋯ |
| L(s) = 1 | + (−0.485 − 0.312i)3-s + (−0.0815 + 0.178i)5-s + (−0.125 − 0.0367i)7-s + (0.138 + 0.303i)9-s + (−1.14 + 1.31i)11-s + (−0.319 + 0.0938i)13-s + (0.0953 − 0.0613i)15-s + (−0.230 − 1.60i)17-s + (−0.148 + 1.03i)19-s + (0.0493 + 0.0569i)21-s + (−0.877 + 0.480i)23-s + (0.629 + 0.726i)25-s + (0.0273 − 0.190i)27-s + (0.282 + 1.96i)29-s + (−0.125 + 0.0808i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.279356 + 0.477818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.279356 + 0.477818i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (4.20 - 2.30i)T \) |
| good | 5 | \( 1 + (0.182 - 0.399i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (0.331 + 0.0973i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (3.78 - 4.37i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (1.15 - 0.338i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.949 + 6.60i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.647 - 4.50i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.52 - 10.5i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (0.700 - 0.450i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (1.33 + 2.92i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (4.14 - 9.08i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.79 + 1.15i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + (-6.03 - 1.77i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (0.789 - 0.231i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-8.65 + 5.56i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-8.35 - 9.63i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (5.85 + 6.75i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.436 + 3.03i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-10.0 + 2.94i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.05 + 2.31i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (3.64 + 2.34i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (4.84 - 10.6i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.10316081082812987063053047552, −10.14984072849193748567678689106, −9.592231061862342071706694222061, −8.239100069405139526714132858839, −7.30226419754815901982157125123, −6.79013433421575023896730265027, −5.34504838263058719285066135251, −4.78533224196603706750083021198, −3.18162386305601440887297373855, −1.84981438556257671287314431557,
0.32080948511047977587697520401, 2.46866864233287889385008469127, 3.80904138307214018575291539934, 4.89893276716548202416797744822, 5.88007025426371703270271577921, 6.61804421898074460838797458806, 8.133551097101890037985227266756, 8.486270509388668894663853389345, 9.850620293582739993940272246441, 10.53602699530267332577482657977
