L(s) = 1 | + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (2.09 + 3.62i)5-s + 3.18·7-s − 7.99·8-s + (−4.18 + 7.25i)10-s − 69.4·11-s + (4.06 − 7.03i)13-s + (3.18 + 5.52i)14-s + (−8 − 13.8i)16-s + (−53.0 − 91.8i)17-s + (42.6 + 70.9i)19-s − 16.7·20-s + (−69.4 − 120. i)22-s + (−88.2 + 152. i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.187 + 0.324i)5-s + 0.172·7-s − 0.353·8-s + (−0.132 + 0.229i)10-s − 1.90·11-s + (0.0867 − 0.150i)13-s + (0.0608 + 0.105i)14-s + (−0.125 − 0.216i)16-s + (−0.756 − 1.31i)17-s + (0.515 + 0.857i)19-s − 0.187·20-s + (−0.672 − 1.16i)22-s + (−0.800 + 1.38i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1796972533\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1796972533\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-42.6 - 70.9i)T \) |
good | 5 | \( 1 + (-2.09 - 3.62i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 - 3.18T + 343T^{2} \) |
| 11 | \( 1 + 69.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-4.06 + 7.03i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (53.0 + 91.8i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 23 | \( 1 + (88.2 - 152. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (33.1 - 57.3i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (207. + 359. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (57.9 + 100. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-310. + 537. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (185. - 322. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-45.8 - 79.4i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (109. - 189. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-72.6 + 125. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-443. - 768. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (99.5 + 172. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-194. - 337. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (212. - 368. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (209. + 363. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.76917299056238083642740803756, −10.69301756352104108770944625102, −9.895541374035683206358440601738, −8.689088912628561285645119815666, −7.69557121719748655253220429301, −7.04400744167764005726699015982, −5.62036726182807570162989298031, −5.08803998349905132002033947847, −3.54769411364595463017613845892, −2.31200060207051063190086304675,
0.05162664695882702425637287273, 1.84650258791015523442720045075, 2.99669298242396030969692466522, 4.48391680781548317038092242385, 5.28492461426173211446214832597, 6.41043967445246368256443448616, 7.83555697025880868609347169445, 8.670369306494728716451130119572, 9.790349375866149488933067003872, 10.70138891939907576613352734637