| L(s) = 1 | + (0.366 − 1.36i)2-s + (−0.307 + 1.14i)3-s + (−1.73 − i)4-s + (0.471 + 4.97i)5-s + (1.45 + 0.841i)6-s + (6.02 − 1.61i)7-s + (−2 + 1.99i)8-s + (6.56 + 3.79i)9-s + (6.97 + 1.17i)10-s + (3.84 − 2.21i)11-s + (1.68 − 1.68i)12-s + (9.21 + 9.16i)13-s − 8.82i·14-s + (−5.86 − 0.991i)15-s + (1.99 + 3.46i)16-s + (−2.88 − 10.7i)17-s + ⋯ |
| L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.102 + 0.382i)3-s + (−0.433 − 0.250i)4-s + (0.0942 + 0.995i)5-s + (0.242 + 0.140i)6-s + (0.861 − 0.230i)7-s + (−0.250 + 0.249i)8-s + (0.729 + 0.421i)9-s + (0.697 + 0.117i)10-s + (0.349 − 0.201i)11-s + (0.140 − 0.140i)12-s + (0.708 + 0.705i)13-s − 0.630i·14-s + (−0.390 − 0.0660i)15-s + (0.124 + 0.216i)16-s + (−0.169 − 0.632i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0540i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61930 + 0.0437588i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61930 + 0.0437588i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.366 + 1.36i)T \) |
| 5 | \( 1 + (-0.471 - 4.97i)T \) |
| 13 | \( 1 + (-9.21 - 9.16i)T \) |
| good | 3 | \( 1 + (0.307 - 1.14i)T + (-7.79 - 4.5i)T^{2} \) |
| 7 | \( 1 + (-6.02 + 1.61i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (-3.84 + 2.21i)T + (60.5 - 104. i)T^{2} \) |
| 17 | \( 1 + (2.88 + 10.7i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (4.90 - 8.50i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-3.92 + 14.6i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + (9.23 - 5.33i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + 8.66iT - 961T^{2} \) |
| 37 | \( 1 + (-6.31 + 23.5i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (50.3 - 29.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-65.8 + 17.6i)T + (1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (34.2 + 34.2i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (49.4 + 49.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-3.03 + 5.26i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (28.9 - 50.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-15.5 + 58.0i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-8.23 - 4.75i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-84.9 + 84.9i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 119. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-41.0 + 41.0i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (61.4 + 106. i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-13.2 + 3.55i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−13.16617793110894360640174375347, −11.71525226670712357438577989410, −11.00982605236221215820460951504, −10.30636358796928873046931294856, −9.148419052526745522725654368795, −7.71235744088672803588546954982, −6.41228035427913254329233232384, −4.80564608283120148187393928511, −3.68521786954187531802923633572, −1.88654839823809815191501617814,
1.35537153048057960177737636723, 4.07086221183972843918079687265, 5.23381595832567995170140419049, 6.39540690917935926908585052212, 7.74834799752711558297901041859, 8.586481143918785232262778482561, 9.647232571208853634112117114077, 11.20332006797922139306041909642, 12.40765924558135245203058921407, 13.00110232740179293172832458172
