L(s) = 1 | + (0.347 − 0.601i)5-s − 0.305·7-s + 4.82·11-s + (−0.5 − 0.866i)13-s + (−3.75 + 6.51i)17-s + (−3.06 − 3.10i)19-s + (−0.347 − 0.601i)23-s + (2.25 + 3.91i)25-s + (5.06 + 8.77i)29-s + 1.82·31-s + (−0.106 + 0.183i)35-s + 6.51·37-s + (2.69 − 4.66i)41-s + (1.84 − 3.19i)43-s + (3 + 5.19i)47-s + ⋯ |
L(s) = 1 | + (0.155 − 0.269i)5-s − 0.115·7-s + 1.45·11-s + (−0.138 − 0.240i)13-s + (−0.911 + 1.57i)17-s + (−0.702 − 0.711i)19-s + (−0.0724 − 0.125i)23-s + (0.451 + 0.782i)25-s + (0.940 + 1.62i)29-s + 0.327·31-s + (−0.0179 + 0.0310i)35-s + 1.07·37-s + (0.420 − 0.728i)41-s + (0.281 − 0.487i)43-s + (0.437 + 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.896962598\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.896962598\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.06 + 3.10i)T \) |
good | 5 | \( 1 + (-0.347 + 0.601i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 0.305T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.75 - 6.51i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.347 + 0.601i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.06 - 8.77i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 - 6.51T + 37T^{2} \) |
| 41 | \( 1 + (-2.69 + 4.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.84 + 3.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.71 - 4.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 3.53i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.194 + 0.337i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.91 + 6.77i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.45 - 9.44i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.19 + 3.80i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.21 + 12.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 0.739T + 83T^{2} \) |
| 89 | \( 1 + (0.411 + 0.712i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.45 - 9.44i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.937116292485194554565635758932, −8.342198997544243463694547239671, −7.24445597065808633104255601387, −6.50356432728979731524416388824, −6.02037836049177771536113894732, −4.83053553113353994553425076564, −4.18999048093547726502861855378, −3.28493329397287926861414625354, −2.06881499170328886235802354620, −1.06094626401486641324079610863,
0.74094493600358410514720935251, 2.11939218139322170804021591130, 2.94904551716924765285441746746, 4.21436217504602122823624421724, 4.56493046952864822778998190212, 5.93061411438753502572946574465, 6.50162531582658535402193874277, 7.06371522310876048890702052123, 8.095836579006182684552437166374, 8.793773937512428548396121437050