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
−10.11241744816861088708714019651, −9.306046006880526895785787988889, −8.268851965039353195859662112232, −7.46337477731742187121379879684, −6.36335678553221728155385228183, −5.47279731803433249210202544336, −4.65799353030933387525403268764, −3.97972261891600432274554994539, −2.73266986618539166654942936528, −1.45305104885622600749290826195,
1.18490471125347180775511616800, 2.45169076014316144687351620901, 3.68625481891426199239252779590, 4.70947562530188867873730878826, 5.62646678368119320804439618350, 6.30513991186625009997721914403, 7.43441432822059744863391625857, 7.906881877006796086585739890636, 8.824487513536828351607234639957, 9.927673329867766761896832208509