| L(s) = 1 | + (−16 + 27.7i)2-s + (321. + 271. i)3-s + (−511. − 886. i)4-s + (459. + 795. i)5-s + (−1.26e4 + 4.58e3i)6-s + (−1.92e4 + 3.33e4i)7-s + 3.27e4·8-s + (3.01e4 + 1.74e5i)9-s − 2.93e4·10-s + (−4.74e4 + 8.22e4i)11-s + (7.55e4 − 4.24e5i)12-s + (1.74e5 + 3.01e5i)13-s + (−6.16e5 − 1.06e6i)14-s + (−6.78e4 + 3.80e5i)15-s + (−5.24e5 + 9.08e5i)16-s − 1.03e7·17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.764 + 0.644i)3-s + (−0.249 − 0.433i)4-s + (0.0657 + 0.113i)5-s + (−0.664 + 0.240i)6-s + (−0.433 + 0.750i)7-s + 0.353·8-s + (0.170 + 0.985i)9-s − 0.0929·10-s + (−0.0888 + 0.153i)11-s + (0.0876 − 0.492i)12-s + (0.130 + 0.225i)13-s + (−0.306 − 0.530i)14-s + (−0.0230 + 0.129i)15-s + (−0.125 + 0.216i)16-s − 1.76·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00351i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.00214190 + 1.21822i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00214190 + 1.21822i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (16 - 27.7i)T \) |
| 3 | \( 1 + (-321. - 271. i)T \) |
| good | 5 | \( 1 + (-459. - 795. i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 7 | \( 1 + (1.92e4 - 3.33e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (4.74e4 - 8.22e4i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + (-1.74e5 - 3.01e5i)T + (-8.96e11 + 1.55e12i)T^{2} \) |
| 17 | \( 1 + 1.03e7T + 3.42e13T^{2} \) |
| 19 | \( 1 + 9.58e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + (6.91e6 + 1.19e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (3.40e7 - 5.89e7i)T + (-6.10e15 - 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-7.74e7 - 1.34e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 - 4.54e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (-2.06e8 - 3.58e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-9.65e7 + 1.67e8i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + (1.40e9 - 2.43e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + 7.13e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-4.00e9 - 6.93e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-3.66e9 + 6.34e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-4.41e9 - 7.64e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 5.49e8T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.48e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + (-1.45e10 + 2.51e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + (-2.16e10 + 3.74e10i)T + (-6.43e20 - 1.11e21i)T^{2} \) |
| 89 | \( 1 + 6.18e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (6.81e10 - 1.18e11i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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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
−16.31518639505608376712230162135, −15.45302502135774132820675147418, −14.41385360215767861752201208933, −12.97582796577766450131058322200, −10.78152773535418866395504133950, −9.341416109472665431104733432275, −8.388866811448916088097142320435, −6.48762397628948239402137431716, −4.53327685938179558627479026366, −2.42559915825558859574889212728,
0.49717583515892689962241238609, 2.21825327881399161694201537059, 3.89521454465062672682518350727, 6.76826606989768502245383351338, 8.283271378453107200444932649860, 9.587558018736389416994378556177, 11.17222716043380928805603721424, 12.91893398253999923794218847247, 13.55388884618097513225592458101, 15.24427976276413764417625026491
