L(s) = 1 | + (0.876 − 1.11i)2-s + (−0.817 − 1.52i)3-s + (−0.465 − 1.94i)4-s + (0.00150 + 0.00141i)5-s + (−2.41 − 0.430i)6-s + (−0.227 − 0.0266i)7-s + (−2.56 − 1.18i)8-s + (−1.66 + 2.49i)9-s + (0.00289 − 0.000426i)10-s + (−1.22 − 0.368i)11-s + (−2.59 + 2.30i)12-s + (−4.36 − 0.254i)13-s + (−0.229 + 0.229i)14-s + (0.000937 − 0.00345i)15-s + (−3.56 + 1.80i)16-s + (−3.40 + 1.24i)17-s + ⋯ |
L(s) = 1 | + (0.619 − 0.785i)2-s + (−0.471 − 0.881i)3-s + (−0.232 − 0.972i)4-s + (0.000672 + 0.000634i)5-s + (−0.984 − 0.175i)6-s + (−0.0860 − 0.0100i)7-s + (−0.907 − 0.419i)8-s + (−0.554 + 0.832i)9-s + (0.000915 − 0.000134i)10-s + (−0.370 − 0.111i)11-s + (−0.747 + 0.664i)12-s + (−1.20 − 0.0704i)13-s + (−0.0612 + 0.0613i)14-s + (0.000242 − 0.000892i)15-s + (−0.891 + 0.452i)16-s + (−0.826 + 0.300i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.339346 + 0.766920i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.339346 + 0.766920i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.876 + 1.11i)T \) |
| 3 | \( 1 + (0.817 + 1.52i)T \) |
good | 5 | \( 1 + (-0.00150 - 0.00141i)T + (0.290 + 4.99i)T^{2} \) |
| 7 | \( 1 + (0.227 + 0.0266i)T + (6.81 + 1.61i)T^{2} \) |
| 11 | \( 1 + (1.22 + 0.368i)T + (9.19 + 6.04i)T^{2} \) |
| 13 | \( 1 + (4.36 + 0.254i)T + (12.9 + 1.50i)T^{2} \) |
| 17 | \( 1 + (3.40 - 1.24i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.999 + 2.74i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-3.60 + 0.421i)T + (22.3 - 5.30i)T^{2} \) |
| 29 | \( 1 + (-1.10 + 2.20i)T + (-17.3 - 23.2i)T^{2} \) |
| 31 | \( 1 + (0.0468 + 0.0629i)T + (-8.89 + 29.6i)T^{2} \) |
| 37 | \( 1 + (0.538 - 0.641i)T + (-6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (5.74 + 3.77i)T + (16.2 + 37.6i)T^{2} \) |
| 43 | \( 1 + (-0.284 - 1.20i)T + (-38.4 + 19.2i)T^{2} \) |
| 47 | \( 1 + (0.816 - 1.09i)T + (-13.4 - 45.0i)T^{2} \) |
| 53 | \( 1 + (-7.12 - 4.11i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.96 + 2.08i)T + (49.2 - 32.4i)T^{2} \) |
| 61 | \( 1 + (0.693 - 0.299i)T + (41.8 - 44.3i)T^{2} \) |
| 67 | \( 1 + (5.83 + 11.6i)T + (-40.0 + 53.7i)T^{2} \) |
| 71 | \( 1 + (-0.402 + 2.28i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.30 + 7.41i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-9.43 + 6.20i)T + (31.2 - 72.5i)T^{2} \) |
| 83 | \( 1 + (-0.822 - 1.25i)T + (-32.8 + 76.2i)T^{2} \) |
| 89 | \( 1 + (-0.854 - 4.84i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (5.49 + 5.82i)T + (-5.64 + 96.8i)T^{2} \) |
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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
−10.35371779269158304955881941535, −9.317391843128243927056832591436, −8.255302640542933555414451594544, −7.08427027311933964247263914004, −6.35191828767824969668307750584, −5.26324276916823801910664447187, −4.55833560868218154343104726925, −2.92539788111550685725978776423, −2.03878502833286322315649917408, −0.36779483996049428331819634944,
2.76959165628905205620307489948, 3.86722738240065589774642789950, 4.94109526594303235107952549083, 5.38997595113617834315065627701, 6.60495417026946271434966876797, 7.33609297179727038655236937619, 8.502022620729246999261108469780, 9.353705177857470326529965860858, 10.13639900961713214101581152368, 11.25575603051998823178670132617