L(s) = 1 | − 0.226·2-s + 3-s − 1.94·4-s + (−0.777 − 2.09i)5-s − 0.226·6-s + (0.735 − 2.54i)7-s + 0.894·8-s + 9-s + (0.176 + 0.475i)10-s + (2.49 − 2.19i)11-s − 1.94·12-s + 3.41i·13-s + (−0.166 + 0.576i)14-s + (−0.777 − 2.09i)15-s + 3.69·16-s − 1.69i·17-s + ⋯ |
L(s) = 1 | − 0.160·2-s + 0.577·3-s − 0.974·4-s + (−0.347 − 0.937i)5-s − 0.0925·6-s + (0.277 − 0.960i)7-s + 0.316·8-s + 0.333·9-s + (0.0557 + 0.150i)10-s + (0.750 − 0.660i)11-s − 0.562·12-s + 0.948i·13-s + (−0.0445 + 0.153i)14-s + (−0.200 − 0.541i)15-s + 0.923·16-s − 0.410i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.059082493\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.059082493\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + (0.777 + 2.09i)T \) |
| 7 | \( 1 + (-0.735 + 2.54i)T \) |
| 11 | \( 1 + (-2.49 + 2.19i)T \) |
good | 2 | \( 1 + 0.226T + 2T^{2} \) |
| 13 | \( 1 - 3.41iT - 13T^{2} \) |
| 17 | \( 1 + 1.69iT - 17T^{2} \) |
| 19 | \( 1 + 4.66T + 19T^{2} \) |
| 23 | \( 1 + 3.95iT - 23T^{2} \) |
| 29 | \( 1 - 3.66iT - 29T^{2} \) |
| 31 | \( 1 + 8.62iT - 31T^{2} \) |
| 37 | \( 1 + 3.09iT - 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 0.542T + 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 - 6.65iT - 53T^{2} \) |
| 59 | \( 1 + 5.11iT - 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 8.03iT - 67T^{2} \) |
| 71 | \( 1 + 3.92T + 71T^{2} \) |
| 73 | \( 1 + 0.227iT - 73T^{2} \) |
| 79 | \( 1 - 9.93iT - 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 + 4.81iT - 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−9.136188022961021890829683825229, −8.839523857048665361752804244095, −8.069517505644670696252284800534, −7.26192737615664884537786033468, −6.13685958217452126013237952869, −4.76681676656089289177828414113, −4.28026774158452474618815837889, −3.57452303962226877160719172405, −1.68689676661655259389353197366, −0.48348256008960172629515311757,
1.73918912466217670230544187627, 3.01437761254423579430679313971, 3.87028217019915803654862541336, 4.84251097681655085332672249490, 5.89641070808553534627792180053, 6.90248646911760264517914077327, 7.87945275529224919128706584381, 8.492265826161603332242576211329, 9.152696805802678884840232032669, 10.08764226304369259532289528625