| L(s) = 1 | + 1.41i·2-s + (−2.46 − 1.70i)3-s − 2.00·4-s + (2.41 − 3.48i)6-s − 2.64·7-s − 2.82i·8-s + (3.15 + 8.42i)9-s + 2.66i·11-s + (4.93 + 3.41i)12-s + 5.56·13-s − 3.74i·14-s + 4.00·16-s − 18.5i·17-s + (−11.9 + 4.46i)18-s − 0.634·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.821 − 0.569i)3-s − 0.500·4-s + (0.402 − 0.581i)6-s − 0.377·7-s − 0.353i·8-s + (0.350 + 0.936i)9-s + 0.242i·11-s + (0.410 + 0.284i)12-s + 0.427·13-s − 0.267i·14-s + 0.250·16-s − 1.09i·17-s + (−0.662 + 0.248i)18-s − 0.0334·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2567289965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2567289965\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (2.46 + 1.70i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 11 | \( 1 - 2.66iT - 121T^{2} \) |
| 13 | \( 1 - 5.56T + 169T^{2} \) |
| 17 | \( 1 + 18.5iT - 289T^{2} \) |
| 19 | \( 1 + 0.634T + 361T^{2} \) |
| 23 | \( 1 - 15.6iT - 529T^{2} \) |
| 29 | \( 1 - 43.3iT - 841T^{2} \) |
| 31 | \( 1 - 13.5T + 961T^{2} \) |
| 37 | \( 1 + 35.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 15.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 64.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 52.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 51.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 32.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 104.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 113.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 36.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 144.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 57.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 45.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 80.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 96.2T + 9.40e3T^{2} \) |
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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
−9.344194301402840365068768641066, −8.493753896389366252124908626851, −7.38722736374602377748253114056, −7.00836489322940191149667930697, −6.05910256735177285979191281416, −5.32598333496649503715631540948, −4.46873094901449364921569297729, −3.09684167004689861298138966689, −1.48267898968041601208883453584, −0.10073299613356455525567657452,
1.22818858274992730409531432507, 2.79333082906353530752914376356, 3.93400081845314786276320355179, 4.51904675290480177645537106559, 5.84799878901932131172382109519, 6.21578602484559248484159621317, 7.54844123170706152279719260611, 8.691154940296191124076227323143, 9.320871524082514149292749408101, 10.40901115759326911660378748911
