L(s) = 1 | − 2-s + (0.5 + 0.866i)3-s + 4-s + (−1 + 1.73i)5-s + (−0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.499 + 0.866i)9-s + (1 − 1.73i)10-s + (−1.5 + 2.59i)11-s + (0.5 + 0.866i)12-s + (−0.5 − 0.866i)14-s − 1.99·15-s + 16-s + (1 + 1.73i)17-s + (0.499 − 0.866i)18-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (−0.447 + 0.774i)5-s + (−0.204 − 0.353i)6-s + (0.188 + 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.316 − 0.547i)10-s + (−0.452 + 0.783i)11-s + (0.144 + 0.249i)12-s + (−0.133 − 0.231i)14-s − 0.516·15-s + 0.250·16-s + (0.242 + 0.420i)17-s + (0.117 − 0.204i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0336 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0336 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.575023 + 0.594737i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.575023 + 0.594737i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-5.5 + 0.866i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 37 | \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6 + 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4 + 6.92i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.5 - 9.52i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−12.61945966651026058468434914468, −11.64235420535508971106393877351, −10.64145535038080118737115280129, −9.990397400618930634315670827914, −8.848844777725781948786964654230, −7.83839829602055826644225828930, −6.94894955269803767596344740780, −5.43947792065295783103648066226, −3.80467402775441427998292768060, −2.38137687018075358794186271103,
0.955702912291211768812930275747, 2.96324101856691697297794933199, 4.73654110686147613734853210494, 6.24961365787797431802097801420, 7.55071717310242492544662252147, 8.264257156864171869084487029135, 9.098281142134381652068806066735, 10.31240066236971583950187066428, 11.42377480104219494118947323786, 12.21763544383976286729633957311