| L(s) = 1 | + (3.53 + 2.04i)2-s + (−2.57 + 4.51i)3-s + (4.33 + 7.51i)4-s + (−2.5 + 4.33i)5-s + (−18.3 + 10.7i)6-s + (0.627 + 18.5i)7-s + 2.75i·8-s + (−13.7 − 23.2i)9-s + (−17.6 + 10.2i)10-s + (−7.33 + 4.23i)11-s + (−45.0 + 0.236i)12-s + 5.50i·13-s + (−35.5 + 66.7i)14-s + (−13.1 − 22.4i)15-s + (29.0 − 50.3i)16-s + (60.2 + 104. i)17-s + ⋯ |
| L(s) = 1 | + (1.25 + 0.721i)2-s + (−0.495 + 0.868i)3-s + (0.542 + 0.938i)4-s + (−0.223 + 0.387i)5-s + (−1.24 + 0.728i)6-s + (0.0338 + 0.999i)7-s + 0.121i·8-s + (−0.509 − 0.860i)9-s + (−0.559 + 0.322i)10-s + (−0.200 + 0.116i)11-s + (−1.08 + 0.00568i)12-s + 0.117i·13-s + (−0.679 + 1.27i)14-s + (−0.225 − 0.386i)15-s + (0.454 − 0.786i)16-s + (0.860 + 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.573i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.818 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.692113 + 2.19379i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.692113 + 2.19379i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.57 - 4.51i)T \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
| 7 | \( 1 + (-0.627 - 18.5i)T \) |
| good | 2 | \( 1 + (-3.53 - 2.04i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (7.33 - 4.23i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 5.50iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-60.2 - 104. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-23.3 - 13.5i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (2.98 + 1.72i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 72.7iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-242. + 139. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-44.8 + 77.7i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 221.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 495.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (100. - 173. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (63.3 - 36.6i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-180. - 312. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (586. + 338. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (385. + 668. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 801. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-881. + 509. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-96.8 + 167. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (310. - 538. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 30.4iT - 9.12e5T^{2} \) |
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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
−14.02662417340664973817446191249, −12.57223643679329540579240302851, −11.96608076721174184927536140390, −10.66760220658747276381851073636, −9.504161456575236801911759494221, −7.977654021766247819935131409481, −6.32778974532595850929509977801, −5.65131749810295110752002030500, −4.45214259641892685842251668420, −3.22146426263706670445063777858,
1.00074380248390908147649060169, 2.91477320823091963292292248179, 4.53662576809045237713001353593, 5.54244273035957823693046162552, 7.03470074201040815275767069620, 8.153236921108248750917064862299, 10.12977803831734715724591085258, 11.29507211275700081128750927692, 11.94810108095110708500304674751, 12.85806399340725586783120712354
