| L(s) = 1 | + (−1.05 + 1.37i)3-s + (−0.533 + 3.71i)5-s + (0.465 + 4.87i)7-s + (−0.755 − 2.90i)9-s + (−4.53 + 1.81i)11-s + (3.30 + 3.46i)13-s + (−4.52 − 4.66i)15-s + (5.33 − 1.84i)17-s + (−0.797 − 0.0761i)19-s + (−7.16 − 4.52i)21-s + (2.77 − 0.132i)23-s + (−8.69 − 2.55i)25-s + (4.77 + 2.03i)27-s + (0.708 − 0.408i)29-s + (5.31 − 5.57i)31-s + ⋯ |
| L(s) = 1 | + (−0.611 + 0.791i)3-s + (−0.238 + 1.65i)5-s + (0.175 + 1.84i)7-s + (−0.251 − 0.967i)9-s + (−1.36 + 0.547i)11-s + (0.917 + 0.962i)13-s + (−1.16 − 1.20i)15-s + (1.29 − 0.448i)17-s + (−0.183 − 0.0174i)19-s + (−1.56 − 0.986i)21-s + (0.578 − 0.0275i)23-s + (−1.73 − 0.510i)25-s + (0.919 + 0.392i)27-s + (0.131 − 0.0759i)29-s + (0.954 − 1.00i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.104948 - 1.07399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.104948 - 1.07399i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.05 - 1.37i)T \) |
| 67 | \( 1 + (8.18 + 0.122i)T \) |
| good | 5 | \( 1 + (0.533 - 3.71i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.465 - 4.87i)T + (-6.87 + 1.32i)T^{2} \) |
| 11 | \( 1 + (4.53 - 1.81i)T + (7.96 - 7.59i)T^{2} \) |
| 13 | \( 1 + (-3.30 - 3.46i)T + (-0.618 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-5.33 + 1.84i)T + (13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (0.797 + 0.0761i)T + (18.6 + 3.59i)T^{2} \) |
| 23 | \( 1 + (-2.77 + 0.132i)T + (22.8 - 2.18i)T^{2} \) |
| 29 | \( 1 + (-0.708 + 0.408i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.31 + 5.57i)T + (-1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-2.23 + 3.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.67 - 1.47i)T + (38.0 + 15.2i)T^{2} \) |
| 43 | \( 1 + (-0.934 - 0.809i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-0.757 + 1.46i)T + (-27.2 - 38.2i)T^{2} \) |
| 53 | \( 1 + (1.96 + 2.27i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.87 - 9.78i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (1.09 - 2.74i)T + (-44.1 - 42.0i)T^{2} \) |
| 71 | \( 1 + (0.732 + 0.253i)T + (55.8 + 43.8i)T^{2} \) |
| 73 | \( 1 + (-7.12 - 2.85i)T + (52.8 + 50.3i)T^{2} \) |
| 79 | \( 1 + (8.83 - 2.14i)T + (70.2 - 36.1i)T^{2} \) |
| 83 | \( 1 + (1.89 - 2.41i)T + (-19.5 - 80.6i)T^{2} \) |
| 89 | \( 1 + (-2.68 + 4.18i)T + (-36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (3.01 + 1.74i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.76337305850876826890734260929, −9.967967017141977224608844387054, −9.242059382126518811518429804261, −8.176098940765660074050438143080, −7.16754086652105196746361107703, −6.06648866277216273785231839755, −5.64712162654484799519512025097, −4.45204710291995113883062298837, −3.11779319834867489389020565463, −2.44938993469895920665986662768,
0.69375340986680012338671780279, 1.19276741089545869069578555363, 3.33803695873598765260210945119, 4.60971967881594175578677564743, 5.28201771695107139595484242919, 6.18211028203337155626737375590, 7.57982274989501420615604426198, 7.925470763492094873488113163149, 8.580925670516503833899139788722, 10.14489497087451943039974160062
