| L(s) = 1 | + (−1.93 + 1.11i)5-s + (−0.973 + 1.68i)7-s + (−13.6 − 7.90i)11-s + (11.4 + 19.8i)13-s − 21.9i·17-s + 16.8·19-s + (24.2 − 13.9i)23-s + (2.5 − 4.33i)25-s + (36.0 + 20.8i)29-s + (−25.9 − 45.0i)31-s − 4.35i·35-s − 49.8·37-s + (−52.0 + 30.0i)41-s + (−25.1 + 43.5i)43-s + (61.6 + 35.5i)47-s + ⋯ |
| L(s) = 1 | + (−0.387 + 0.223i)5-s + (−0.139 + 0.240i)7-s + (−1.24 − 0.719i)11-s + (0.882 + 1.52i)13-s − 1.29i·17-s + 0.885·19-s + (1.05 − 0.608i)23-s + (0.100 − 0.173i)25-s + (1.24 + 0.717i)29-s + (−0.838 − 1.45i)31-s − 0.124i·35-s − 1.34·37-s + (−1.26 + 0.733i)41-s + (−0.584 + 1.01i)43-s + (1.31 + 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.209724435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209724435\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
| good | 7 | \( 1 + (0.973 - 1.68i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (13.6 + 7.90i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-11.4 - 19.8i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 21.9iT - 289T^{2} \) |
| 19 | \( 1 - 16.8T + 361T^{2} \) |
| 23 | \( 1 + (-24.2 + 13.9i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-36.0 - 20.8i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (25.9 + 45.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 49.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + (52.0 - 30.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (25.1 - 43.5i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-61.6 - 35.5i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 21.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (34.4 - 19.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (53.7 - 93.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-40.9 - 70.9i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 62.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + (0.00315 - 0.00546i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (55.5 + 32.0i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 51.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (22.3 - 38.6i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.176307185511018230071329776428, −8.793764835103565592246514869034, −7.76415498285784351919535096505, −7.06368922287717576381287863665, −6.24686557164737761056282656900, −5.26406787803515859021326334577, −4.48742778241567324448193975485, −3.28056179251286964301382952927, −2.60988182757519215155742439004, −1.06314895886299704225333495793,
0.37746234404241738156905309443, 1.67708066489604658645991432791, 3.13520205734697234660909364671, 3.70381334305155774150996277539, 5.16726378335885947616401268581, 5.38417369413886384696765904632, 6.72693060709548289072503076668, 7.46663474022009244992101486544, 8.265767785778568303175066386725, 8.740005057300420557863192940782
