L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.5 + 2.59i)5-s + (−0.5 − 2.59i)7-s − 0.999·8-s + (−1.5 + 2.59i)10-s + (−1.5 + 2.59i)11-s + 2·13-s + (2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (−1 − 1.73i)19-s − 3·20-s − 3·22-s + (−3 − 5.19i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.188 − 0.981i)7-s − 0.353·8-s + (−0.474 + 0.821i)10-s + (−0.452 + 0.783i)11-s + 0.554·13-s + (0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (−0.229 − 0.397i)19-s − 0.670·20-s − 0.639·22-s + (−0.625 − 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08261 + 0.720138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08261 + 0.720138i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.73308240383005588446534852794, −12.94644669985643180175607421746, −11.45648750120319893430667875636, −10.35127385019311216336548407302, −9.576763478502629146939969152428, −7.78538308212035016131453949842, −6.96692914545533405908879123900, −5.98856998440022256932389911984, −4.40605070508217192420908716807, −2.81590849237935361042594327890,
1.76705273127223839865729827093, 3.64567419209290076682487171822, 5.44853858631301919043361281418, 5.87236766799343943785364068110, 8.228061830397331752145247447386, 9.038753596637336024596476602616, 10.04429345795462731799351112682, 11.27416781180045924414634966393, 12.42095484519876077444287365877, 12.97452706875068129200745924857