| L(s) = 1 | + (−0.575 − 0.686i)2-s + (1.25 − 1.18i)3-s + (0.207 − 1.17i)4-s + (0.100 + 0.0844i)5-s + (−1.54 − 0.178i)6-s + (−1.70 − 2.02i)7-s + (−2.48 + 1.43i)8-s + (0.169 − 2.99i)9-s − 0.117i·10-s + (2.57 + 3.06i)11-s + (−1.14 − 1.73i)12-s + (1.02 + 2.81i)13-s + (−0.409 + 2.33i)14-s + (0.227 − 0.0134i)15-s + (0.160 + 0.0583i)16-s − 0.344·17-s + ⋯ |
| L(s) = 1 | + (−0.407 − 0.485i)2-s + (0.726 − 0.686i)3-s + (0.103 − 0.589i)4-s + (0.0450 + 0.0377i)5-s + (−0.629 − 0.0730i)6-s + (−0.643 − 0.765i)7-s + (−0.877 + 0.506i)8-s + (0.0566 − 0.998i)9-s − 0.0372i·10-s + (0.776 + 0.925i)11-s + (−0.329 − 0.500i)12-s + (0.284 + 0.781i)13-s + (−0.109 + 0.623i)14-s + (0.0586 − 0.00346i)15-s + (0.0401 + 0.0145i)16-s − 0.0835·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.630715 - 0.966676i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.630715 - 0.966676i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.25 + 1.18i)T \) |
| 7 | \( 1 + (1.70 + 2.02i)T \) |
| good | 2 | \( 1 + (0.575 + 0.686i)T + (-0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.100 - 0.0844i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-2.57 - 3.06i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.02 - 2.81i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + 0.344T + 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + (-2.18 - 5.98i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-2.29 + 6.31i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-8.59 - 1.51i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.64 - 6.30i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (9.04 - 3.29i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.350 - 1.98i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.771 - 4.37i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-5.49 + 3.17i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.167 + 0.0608i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (4.60 - 0.811i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.83 - 7.40i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.373 - 0.215i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.31 - 0.761i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.81 + 1.52i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (6.81 + 2.48i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + (12.2 - 2.15i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.08498416413137732434642721855, −11.39587471896365330440212520118, −9.890832164407352185866222030309, −9.561911050458952332325704892092, −8.391130209093511805745055050330, −6.92327675056423041108427965450, −6.41291175955987265072158634096, −4.34484886609598857639151719799, −2.73435395099589523006524410899, −1.25332310184182280441666623128,
2.89025743223912922709913531412, 3.76740496263258838358437162130, 5.61942651120282467337216157097, 6.76891813996214211143850746462, 8.228396275420686036433545818806, 8.684110912560382498824629840213, 9.575473594610449465676754623923, 10.70684621512104008296274313277, 12.00503526314503537260966119774, 12.87838144350133566120987326875
