| L(s) = 1 | + (−0.471 + 1.29i)2-s + (0.0766 + 0.0643i)4-s + (0.459 − 2.60i)5-s + (−1.47 − 2.19i)7-s + (−2.50 + 1.44i)8-s + (3.15 + 1.82i)10-s + (0.681 − 0.120i)11-s + (0.171 + 0.472i)13-s + (3.54 − 0.869i)14-s + (−0.658 − 3.73i)16-s + (2.42 − 4.19i)17-s + (−1.03 + 0.597i)19-s + (0.202 − 0.170i)20-s + (−0.165 + 0.939i)22-s + (4.74 − 5.65i)23-s + ⋯ |
| L(s) = 1 | + (−0.333 + 0.915i)2-s + (0.0383 + 0.0321i)4-s + (0.205 − 1.16i)5-s + (−0.556 − 0.831i)7-s + (−0.886 + 0.511i)8-s + (0.998 + 0.576i)10-s + (0.205 − 0.0362i)11-s + (0.0476 + 0.130i)13-s + (0.946 − 0.232i)14-s + (−0.164 − 0.933i)16-s + (0.587 − 1.01i)17-s + (−0.237 + 0.137i)19-s + (0.0453 − 0.0380i)20-s + (−0.0353 + 0.200i)22-s + (0.990 − 1.18i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14369 - 0.183575i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14369 - 0.183575i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.47 + 2.19i)T \) |
| good | 2 | \( 1 + (0.471 - 1.29i)T + (-1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.459 + 2.60i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.681 + 0.120i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.171 - 0.472i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.42 + 4.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.03 - 0.597i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.74 + 5.65i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.19 + 6.03i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.17 + 6.16i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.71 - 6.43i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.25 - 1.18i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.81 + 10.2i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.93 + 2.45i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (1.13 - 6.41i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.36 - 2.81i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.24 - 1.17i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (0.286 + 0.165i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.13 + 1.81i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.71 - 3.53i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (12.8 + 4.69i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-4.84 - 8.39i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.19 - 0.563i)T + (91.1 - 33.1i)T^{2} \) |
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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
−10.48217009054840939622224903367, −9.549269330895040999858586968797, −8.805497596543486899694217532743, −8.039005076855623436483261976217, −7.08722738342044898413081697900, −6.35696630132789570366505729721, −5.28063691586618896468447107870, −4.26540542781327772117159925497, −2.79147520869428659858466315849, −0.75682802109033268313670598959,
1.64223453434055037304433458889, 2.91810486527921187170044673832, 3.42791706252996699094225457408, 5.38511862460410284470244597014, 6.38535459422415635500134974382, 6.94716131753931408541288795096, 8.425577296517605728706147191820, 9.340527368792353788570938548475, 10.09165034610572026975133324440, 10.75697398979907450353699494187
