| L(s) = 1 | + (−4.03 − 2.33i)2-s + (−5.14 + 0.747i)3-s + (6.87 + 11.9i)4-s + (−12.8 − 3.44i)5-s + (22.5 + 8.97i)6-s + (21.5 − 5.76i)7-s − 26.8i·8-s + (25.8 − 7.68i)9-s + (43.9 + 43.9i)10-s + (10.9 − 2.93i)11-s + (−44.2 − 56.0i)12-s + (8.50 + 14.7i)13-s + (−100. − 26.8i)14-s + (68.7 + 8.11i)15-s + (−7.54 + 13.0i)16-s + (22.8 − 66.2i)17-s + ⋯ |
| L(s) = 1 | + (−1.42 − 0.824i)2-s + (−0.989 + 0.143i)3-s + (0.859 + 1.48i)4-s + (−1.15 − 0.308i)5-s + (1.53 + 0.610i)6-s + (1.16 − 0.311i)7-s − 1.18i·8-s + (0.958 − 0.284i)9-s + (1.38 + 1.38i)10-s + (0.299 − 0.0803i)11-s + (−1.06 − 1.34i)12-s + (0.181 + 0.314i)13-s + (−1.91 − 0.513i)14-s + (1.18 + 0.139i)15-s + (−0.117 + 0.204i)16-s + (0.326 − 0.945i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0791i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00941576 + 0.237417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00941576 + 0.237417i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.14 - 0.747i)T \) |
| 17 | \( 1 + (-22.8 + 66.2i)T \) |
| good | 2 | \( 1 + (4.03 + 2.33i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (12.8 + 3.44i)T + (108. + 62.5i)T^{2} \) |
| 7 | \( 1 + (-21.5 + 5.76i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-10.9 + 2.93i)T + (1.15e3 - 665.5i)T^{2} \) |
| 13 | \( 1 + (-8.50 - 14.7i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 19 | \( 1 + 38.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (51.8 - 193. i)T + (-1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (20.4 + 76.1i)T + (-2.11e4 + 1.21e4i)T^{2} \) |
| 31 | \( 1 + (-64.7 - 17.3i)T + (2.57e4 + 1.48e4i)T^{2} \) |
| 37 | \( 1 + (106. - 106. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (-120. + 448. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (356. + 205. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-50.1 + 86.8i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 31.5iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-538. + 310. i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (375. - 100. i)T + (1.96e5 - 1.13e5i)T^{2} \) |
| 67 | \( 1 + (510. + 884. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (44.0 - 44.0i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (222. - 222. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (-154. + 41.4i)T + (4.26e5 - 2.46e5i)T^{2} \) |
| 83 | \( 1 + (727. + 420. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 908.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (371. + 1.38e3i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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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
−11.65337617934180853942947156025, −11.17617883786486061028341186781, −10.11107009158590445836509998522, −9.015824526473110127329507794419, −7.889813280870591379294033193099, −7.19830246226416958881872658719, −5.14065007884519127079999672636, −3.82826834410099641837277014828, −1.48935989355165523803653256130, −0.24538119308570643825446417720,
1.34520195107530526304325969204, 4.33709428229968821389978095479, 5.85303761014013107656490357097, 6.89777470170018477720776804281, 7.962205549828532024349997317341, 8.412435701631531914850715072717, 10.07249684294203540630653940972, 10.87989287537312686336732444722, 11.64908335026969831782845748682, 12.62587746186450070227365402672
