L(s) = 1 | + (−0.920 − 1.59i)2-s + (−0.695 + 1.20i)4-s − 1.33·5-s + (−2.54 − 0.728i)7-s − 1.12·8-s + (1.22 + 2.12i)10-s − 1.51·11-s + (−2.58 − 4.48i)13-s + (1.17 + 4.72i)14-s + (2.42 + 4.19i)16-s + (−0.774 − 1.34i)17-s + (−1.25 + 2.16i)19-s + (0.927 − 1.60i)20-s + (1.39 + 2.41i)22-s + 7.36·23-s + ⋯ |
L(s) = 1 | + (−0.650 − 1.12i)2-s + (−0.347 + 0.601i)4-s − 0.596·5-s + (−0.961 − 0.275i)7-s − 0.396·8-s + (0.388 + 0.673i)10-s − 0.456·11-s + (−0.717 − 1.24i)13-s + (0.315 + 1.26i)14-s + (0.605 + 1.04i)16-s + (−0.187 − 0.325i)17-s + (−0.287 + 0.497i)19-s + (0.207 − 0.359i)20-s + (0.296 + 0.514i)22-s + 1.53·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0853641 + 0.332237i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0853641 + 0.332237i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.54 + 0.728i)T \) |
good | 2 | \( 1 + (0.920 + 1.59i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 1.33T + 5T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 + (2.58 + 4.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.774 + 1.34i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.25 - 2.16i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.36T + 23T^{2} \) |
| 29 | \( 1 + (-0.0309 + 0.0536i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.92 + 3.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.281 - 0.487i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.51 + 7.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.09 + 8.83i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.75 + 8.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.755 + 1.30i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.22 - 7.31i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.61 + 2.80i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.46 - 6.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + (1.37 + 2.38i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.95 - 5.12i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.80 - 4.85i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.703 - 1.21i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.09 - 10.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.96401365459806877574374087953, −10.80363640161905766818006825701, −10.20022254632082446477539669747, −9.299386362409250385406381334548, −8.161244934194465721823351529513, −7.01930763146187339845100314192, −5.50025440818610696187411449260, −3.68395300331636036103125010239, −2.63473575069796585917130883873, −0.35617371768237503392341351278,
3.00043713575327089756049846821, 4.75273592288863850352114444931, 6.27586416674880033885751495143, 6.98486613380749382289544685881, 7.968868212779612737500920778008, 9.055587215689502072818242161725, 9.690009091391917467720593876787, 11.16749713087238543228671532732, 12.20472897024096506141497040155, 13.10048818085696366466085675311