| L(s) = 1 | + (1.43 + 0.524i)2-s + (0.266 + 0.223i)4-s + (0.560 − 3.17i)5-s + (−0.5 + 0.419i)7-s + (−1.26 − 2.19i)8-s + (2.47 − 4.28i)10-s + (0.173 + 0.984i)11-s + (−2.15 + 0.783i)13-s + (−0.939 + 0.342i)14-s + (−0.794 − 4.50i)16-s + (1.23 − 2.13i)17-s + (−3.35 − 5.81i)19-s + (0.858 − 0.720i)20-s + (−0.266 + 1.50i)22-s + (−0.532 − 0.446i)23-s + ⋯ |
| L(s) = 1 | + (1.01 + 0.370i)2-s + (0.133 + 0.111i)4-s + (0.250 − 1.42i)5-s + (−0.188 + 0.158i)7-s + (−0.447 − 0.775i)8-s + (0.781 − 1.35i)10-s + (0.0523 + 0.296i)11-s + (−0.597 + 0.217i)13-s + (−0.251 + 0.0914i)14-s + (−0.198 − 1.12i)16-s + (0.299 − 0.518i)17-s + (−0.770 − 1.33i)19-s + (0.191 − 0.161i)20-s + (−0.0567 + 0.321i)22-s + (−0.110 − 0.0930i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.51533 - 1.42964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51533 - 1.42964i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-0.173 - 0.984i)T \) |
| good | 2 | \( 1 + (-1.43 - 0.524i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.560 + 3.17i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.419i)T + (1.21 - 6.89i)T^{2} \) |
| 13 | \( 1 + (2.15 - 0.783i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.23 + 2.13i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.35 + 5.81i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.532 + 0.446i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-6.98 - 2.54i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.04 + 0.880i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.22 + 3.85i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.479 + 0.174i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.01 + 11.4i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.91 - 4.96i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 6.30T + 53T^{2} \) |
| 59 | \( 1 + (-0.131 + 0.747i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.34 + 1.13i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.99 + 1.81i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.30 + 9.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.20 - 9.02i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.64 - 3.51i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-9.63 - 3.50i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-9.15 - 15.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.85 - 10.5i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.570064727265495575692988177589, −9.242674713779120990512593752735, −8.313568800569512418546381005567, −7.07430651538893112186733563937, −6.30014357830913640780209201514, −5.10406412574820892284571753165, −4.91855505583232692326586778104, −3.91285161573124189332517347464, −2.45252018598447713354675945143, −0.70681073253747176103853545900,
2.14440538696324461778258151089, 3.10813794959678550828180304918, 3.78727528091814283236452190999, 4.91150509385805070883900407006, 6.06513957449280971813080227258, 6.51366603332464508168717635329, 7.76506458912300572957616405261, 8.499375269270358009430010794234, 9.944130502617866132585960564900, 10.33510333412645462470439077699
