| L(s) = 1 | + (−0.841 − 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (−0.415 + 0.909i)6-s + (2.94 − 3.40i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)10-s + (3.15 − 2.03i)11-s + (0.841 − 0.540i)12-s + (4.55 + 5.25i)13-s + (−4.31 + 1.26i)14-s + (0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (−2.91 + 6.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (−0.0821 − 0.571i)3-s + (0.207 + 0.454i)4-s + (0.429 + 0.125i)5-s + (−0.169 + 0.371i)6-s + (1.11 − 1.28i)7-s + (0.0503 − 0.349i)8-s + (−0.319 + 0.0939i)9-s + (−0.207 − 0.238i)10-s + (0.952 − 0.612i)11-s + (0.242 − 0.156i)12-s + (1.26 + 1.45i)13-s + (−1.15 + 0.338i)14-s + (0.0367 − 0.255i)15-s + (−0.163 + 0.188i)16-s + (−0.705 + 1.54i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16952 - 0.830996i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16952 - 0.830996i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-1.30 + 4.61i)T \) |
| good | 7 | \( 1 + (-2.94 + 3.40i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.15 + 2.03i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-4.55 - 5.25i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.91 - 6.37i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.990 + 2.16i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (2.07 - 4.55i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.564 + 3.92i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-7.21 + 2.11i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (2.11 + 0.620i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.26 - 8.77i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 4.07T + 47T^{2} \) |
| 53 | \( 1 + (-2.56 + 2.96i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (7.13 + 8.23i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.89 + 13.1i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.556i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (4.19 + 2.69i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.59 + 5.67i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (2.60 + 3.00i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (14.6 - 4.29i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.83 - 12.7i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.25 - 1.83i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.62220127729401180487372312283, −9.283162455007401616006655816730, −8.588048791998015489467959939109, −7.87209945336645036604559649138, −6.60934049757860123194814859382, −6.37527644879645320608117468178, −4.49591205284468063325941311461, −3.74480623133783303772988276306, −1.89736107419119100379586489940, −1.18052694952352440299784201512,
1.42982366320279320075127412421, 2.78220194560769187465936990288, 4.41342152199069586716689346988, 5.47541052682930050120246247578, 5.92310395741780259873254454074, 7.25140635684985942300513374175, 8.336561676253428564406318664144, 8.893791281561321798331821990189, 9.578269091398515763497442786662, 10.51371290993556875116995640897
