| L(s) = 1 | + (16 − 27.7i)2-s + (312. − 281. i)3-s + (−511. − 886. i)4-s + (−5.38e3 − 9.31e3i)5-s + (−2.79e3 − 1.31e4i)6-s + (−3.79e4 + 6.57e4i)7-s − 3.27e4·8-s + (1.85e4 − 1.76e5i)9-s − 3.44e5·10-s + (2.11e5 − 3.67e5i)11-s + (−4.09e5 − 1.33e5i)12-s + (5.25e5 + 9.10e5i)13-s + (1.21e6 + 2.10e6i)14-s + (−4.30e6 − 1.40e6i)15-s + (−5.24e5 + 9.08e5i)16-s − 5.13e6·17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.743 − 0.669i)3-s + (−0.249 − 0.433i)4-s + (−0.769 − 1.33i)5-s + (−0.146 − 0.691i)6-s + (−0.853 + 1.47i)7-s − 0.353·8-s + (0.104 − 0.994i)9-s − 1.08·10-s + (0.396 − 0.687i)11-s + (−0.475 − 0.154i)12-s + (0.392 + 0.679i)13-s + (0.603 + 1.04i)14-s + (−1.46 − 0.476i)15-s + (−0.125 + 0.216i)16-s − 0.876·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.203947 + 1.44969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.203947 + 1.44969i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (-312. + 281. i)T \) |
| good | 5 | \( 1 + (5.38e3 + 9.31e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 7 | \( 1 + (3.79e4 - 6.57e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-2.11e5 + 3.67e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + (-5.25e5 - 9.10e5i)T + (-8.96e11 + 1.55e12i)T^{2} \) |
| 17 | \( 1 + 5.13e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.51e5T + 1.16e14T^{2} \) |
| 23 | \( 1 + (2.46e7 + 4.27e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-3.67e7 + 6.37e7i)T + (-6.10e15 - 1.05e16i)T^{2} \) |
| 31 | \( 1 + (5.33e7 + 9.24e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 - 3.65e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (2.89e8 + 5.01e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-7.49e8 + 1.29e9i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + (3.21e8 - 5.56e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 - 3.07e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-3.75e9 - 6.51e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (3.26e9 - 5.65e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (1.00e10 + 1.74e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 3.31e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.42e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + (1.19e10 - 2.07e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + (-2.30e10 + 3.98e10i)T + (-6.43e20 - 1.11e21i)T^{2} \) |
| 89 | \( 1 - 8.40e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-1.25e9 + 2.16e9i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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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
−15.23733696642558557032002185903, −13.57295939052902412766632151402, −12.48123570938989006700092937483, −11.83377729815585993760295039862, −9.120726372826557913484404735542, −8.569465695064109465462914651717, −6.12816246824213540594079388029, −4.02919176586659155206571244871, −2.30109140320900419684370359271, −0.48661527067761370228321582266,
3.24140617888083739964847291980, 4.10986672893826845868963750568, 6.79337076597573871439326031248, 7.76374742217195636269407291953, 9.830831625081113551766750962798, 10.99380252886020940340302508019, 13.25174880497762542181763571456, 14.32816029905141375574962570455, 15.34424293437921218475076465047, 16.24605474640877999091678103550
