L(s) = 1 | − 2-s − 4-s + (−0.5 + 0.866i)5-s + (2 − 1.73i)7-s + 3·8-s + (0.5 − 0.866i)10-s + (2.5 + 4.33i)11-s + (2.5 + 4.33i)13-s + (−2 + 1.73i)14-s − 16-s + (1.5 − 2.59i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (−2.5 − 4.33i)22-s + (1.5 − 2.59i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (0.755 − 0.654i)7-s + 1.06·8-s + (0.158 − 0.273i)10-s + (0.753 + 1.30i)11-s + (0.693 + 1.20i)13-s + (−0.534 + 0.462i)14-s − 0.250·16-s + (0.363 − 0.630i)17-s + (−0.114 − 0.198i)19-s + (0.111 − 0.193i)20-s + (−0.533 − 0.923i)22-s + (0.312 − 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.771514 + 0.187092i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.771514 + 0.187092i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (1.5 - 2.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.56228453862573756095980609702, −11.41339285891468038713768079353, −10.61988458834588128445306079576, −9.527710891089699547149021574599, −8.787510548118890264028300112934, −7.51857919503984356996870283016, −6.85993090869678789422442766272, −4.84405173034031777728197531192, −3.98784000574745962223690651716, −1.54879141719047672260390690944,
1.15954005450070349770990608819, 3.55698322445557741853188652806, 5.01179670474333687540389094978, 6.09809757732958974580992158973, 8.041558117334159433696107494868, 8.355259586042305130754459436618, 9.254913218369034317099183957610, 10.51838487147393649811474419264, 11.32924784759997815795275612755, 12.46212104465039664491971004511