| L(s) = 1 | + (−1.97 − 0.339i)2-s + (2.56 + 1.56i)3-s + (3.76 + 1.33i)4-s + (7.18 − 2.61i)5-s + (−4.51 − 3.94i)6-s + (−5.65 + 0.996i)7-s + (−6.97 − 3.91i)8-s + (4.11 + 8.00i)9-s + (−15.0 + 2.71i)10-s + (2.01 − 5.53i)11-s + (7.56 + 9.31i)12-s + (8.25 + 6.92i)13-s + (11.4 − 0.0456i)14-s + (22.4 + 4.53i)15-s + (12.4 + 10.0i)16-s + (10.4 − 18.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.985 − 0.169i)2-s + (0.853 + 0.521i)3-s + (0.942 + 0.334i)4-s + (1.43 − 0.523i)5-s + (−0.752 − 0.658i)6-s + (−0.807 + 0.142i)7-s + (−0.871 − 0.489i)8-s + (0.457 + 0.889i)9-s + (−1.50 + 0.271i)10-s + (0.183 − 0.503i)11-s + (0.630 + 0.776i)12-s + (0.635 + 0.533i)13-s + (0.819 − 0.00325i)14-s + (1.49 + 0.302i)15-s + (0.776 + 0.630i)16-s + (0.616 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.35118 + 0.113341i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35118 + 0.113341i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.97 + 0.339i)T \) |
| 3 | \( 1 + (-2.56 - 1.56i)T \) |
| good | 5 | \( 1 + (-7.18 + 2.61i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (5.65 - 0.996i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-2.01 + 5.53i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-8.25 - 6.92i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-10.4 + 18.1i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (21.5 - 12.4i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-20.5 - 3.62i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (30.1 - 25.3i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (10.3 + 1.82i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (1.83 - 3.18i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (51.1 + 42.9i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-28.3 + 77.9i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (33.3 - 5.88i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 43.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + (9.37 + 25.7i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (1.47 + 8.39i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (31.0 - 37.0i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-24.2 - 13.9i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-43.5 - 75.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (11.6 + 13.9i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-8.95 - 10.6i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (81.2 + 140. i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-39.6 - 14.4i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−13.48214548849428469196311977631, −12.58743754450351599003085632875, −10.92791149418383277941041372523, −9.920362841985237219806343482895, −9.209925632994540902322964874158, −8.612084852695198279647893347601, −6.92948573830624887748840825531, −5.60591267201087909711656506772, −3.35528467047892510093098286682, −1.84393151510497198679949148028,
1.71558384746853911010381348171, 3.05590041674260721144889609086, 6.11812611559829951766160151373, 6.69075025380542184103206983523, 8.058800819085845684737112270231, 9.268738622617374117556813031377, 9.895759157659611023763306898421, 10.88548424473175971552033713754, 12.73107555788812331277463785031, 13.33485074588853192349947630410
