L(s) = 1 | + (0.965 + 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s − i·6-s + (2.63 − 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (1.17 − 2.04i)11-s + (0.258 − 0.965i)12-s + (−0.0968 + 0.0968i)13-s + (2.61 + 0.431i)14-s + (0.500 + 0.866i)16-s + (3.11 − 0.833i)17-s + (−0.965 + 0.258i)18-s + (−0.434 − 0.752i)19-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s − 0.408i·6-s + (0.995 − 0.0978i)7-s + (0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.355 − 0.615i)11-s + (0.0747 − 0.278i)12-s + (−0.0268 + 0.0268i)13-s + (0.697 + 0.115i)14-s + (0.125 + 0.216i)16-s + (0.754 − 0.202i)17-s + (−0.227 + 0.0610i)18-s + (−0.0996 − 0.172i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.649810616\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.649810616\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.258i)T \) |
good | 11 | \( 1 + (-1.17 + 2.04i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0968 - 0.0968i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.11 + 0.833i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.434 + 0.752i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.833 - 3.11i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 4.08iT - 29T^{2} \) |
| 31 | \( 1 + (0.747 + 0.431i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.51 - 2.54i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 6.86iT - 41T^{2} \) |
| 43 | \( 1 + (-2.57 - 2.57i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.223 + 0.833i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (8.07 - 2.16i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-5.35 + 9.27i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.44 - 4.29i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.28 - 12.2i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 5.13T + 71T^{2} \) |
| 73 | \( 1 + (4.20 + 15.7i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (4.02 - 2.32i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.13 - 7.13i)T - 83iT^{2} \) |
| 89 | \( 1 + (4.98 + 8.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.4 - 11.4i)T + 97iT^{2} \) |
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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
−9.927673329867766761896832208509, −8.824487513536828351607234639957, −7.906881877006796086585739890636, −7.43441432822059744863391625857, −6.30513991186625009997721914403, −5.62646678368119320804439618350, −4.70947562530188867873730878826, −3.68625481891426199239252779590, −2.45169076014316144687351620901, −1.18490471125347180775511616800,
1.45305104885622600749290826195, 2.73266986618539166654942936528, 3.97972261891600432274554994539, 4.65799353030933387525403268764, 5.47279731803433249210202544336, 6.36335678553221728155385228183, 7.46337477731742187121379879684, 8.268851965039353195859662112232, 9.306046006880526895785787988889, 10.11241744816861088708714019651