| 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
−10.51371290993556875116995640897, −9.578269091398515763497442786662, −8.893791281561321798331821990189, −8.336561676253428564406318664144, −7.25140635684985942300513374175, −5.92310395741780259873254454074, −5.47541052682930050120246247578, −4.41342152199069586716689346988, −2.78220194560769187465936990288, −1.42982366320279320075127412421,
1.18052694952352440299784201512, 1.89736107419119100379586489940, 3.74480623133783303772988276306, 4.49591205284468063325941311461, 6.37527644879645320608117468178, 6.60934049757860123194814859382, 7.87209945336645036604559649138, 8.588048791998015489467959939109, 9.283162455007401616006655816730, 10.62220127729401180487372312283
