| L(s) = 1 | + (−0.489 + 1.32i)2-s + (−2.70 − 0.538i)3-s + (−1.51 − 1.30i)4-s + (−2.42 + 1.61i)5-s + (2.04 − 3.32i)6-s + (−0.747 + 1.11i)7-s + (2.46 − 1.37i)8-s + (4.27 + 1.77i)9-s + (−0.960 − 4.00i)10-s + (0.611 + 3.07i)11-s + (3.41 + 4.34i)12-s + (−3.24 − 3.24i)13-s + (−1.11 − 1.53i)14-s + (7.43 − 3.08i)15-s + (0.619 + 3.95i)16-s + (−2.21 + 3.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.346 + 0.938i)2-s + (−1.56 − 0.311i)3-s + (−0.759 − 0.650i)4-s + (−1.08 + 0.724i)5-s + (0.833 − 1.35i)6-s + (−0.282 + 0.422i)7-s + (0.873 − 0.487i)8-s + (1.42 + 0.590i)9-s + (−0.303 − 1.26i)10-s + (0.184 + 0.926i)11-s + (0.986 + 1.25i)12-s + (−0.899 − 0.899i)13-s + (−0.298 − 0.411i)14-s + (1.92 − 0.795i)15-s + (0.154 + 0.987i)16-s + (−0.538 + 0.842i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0138629 - 0.178189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0138629 - 0.178189i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.489 - 1.32i)T \) |
| 17 | \( 1 + (2.21 - 3.47i)T \) |
| good | 3 | \( 1 + (2.70 + 0.538i)T + (2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (2.42 - 1.61i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (0.747 - 1.11i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.611 - 3.07i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (3.24 + 3.24i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.100 + 0.242i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (2.31 - 0.459i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.242 - 0.362i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (-1.10 + 5.55i)T + (-28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (0.474 - 2.38i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.44 - 0.965i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (3.12 - 7.55i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (3.35 - 3.35i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.62 + 1.50i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-2.31 - 0.959i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (4.61 - 6.90i)T + (-23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + (14.4 + 2.88i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-7.57 + 5.06i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (1.02 + 5.14i)T + (-72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (-3.34 + 1.38i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (12.4 - 12.4i)T - 89iT^{2} \) |
| 97 | \( 1 + (3.63 + 5.43i)T + (-37.1 + 89.6i)T^{2} \) |
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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
−15.37687678512626405642474330509, −14.90095216348020185018952044198, −12.94019821704241793681975740817, −12.03291687547191398143516620098, −10.89493866465216745300674346515, −9.863310215969786391239207214029, −7.896847254170810193565159187503, −6.96446095911224085391426977437, −5.94157774086025996692098148793, −4.53215000416354433731754131114,
0.32863637892318942326133747416, 4.01791282461663031544341177355, 5.04534695518303978611805986203, 7.03662754847708411738888351543, 8.654828487828541526573637196399, 9.983533579017987030702143306915, 11.15772882068647526455749294835, 11.80074641422248734339008299802, 12.46443444150786621260993001571, 13.87532674797242960438781460615
