L(s) = 1 | + (−0.207 + 0.358i)3-s + (−0.5 − 0.866i)5-s + (1.41 + 2.44i)9-s + (0.5 − 0.866i)11-s − 2.41·13-s + 0.414·15-s + (−0.207 + 0.358i)17-s + (1 + 1.73i)19-s + (1.12 + 1.94i)23-s + (−0.499 + 0.866i)25-s − 2.41·27-s + 29-s + (0.878 − 1.52i)31-s + (0.207 + 0.358i)33-s + (3.94 + 6.84i)37-s + ⋯ |
L(s) = 1 | + (−0.119 + 0.207i)3-s + (−0.223 − 0.387i)5-s + (0.471 + 0.816i)9-s + (0.150 − 0.261i)11-s − 0.669·13-s + 0.106·15-s + (−0.0502 + 0.0870i)17-s + (0.229 + 0.397i)19-s + (0.233 + 0.404i)23-s + (−0.0999 + 0.173i)25-s − 0.464·27-s + 0.185·29-s + (0.157 − 0.273i)31-s + (0.0360 + 0.0624i)33-s + (0.649 + 1.12i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.326528198\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.326528198\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.207 - 0.358i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 + (0.207 - 0.358i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.12 - 1.94i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + (-0.878 + 1.52i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.94 - 6.84i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 + 0.343T + 43T^{2} \) |
| 47 | \( 1 + (-5.20 - 9.01i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.70 - 8.15i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.12 + 8.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.585 + 1.01i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.707 - 1.22i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + (2.58 - 4.47i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.32 - 12.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + (0.828 + 1.43i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.0710T + 97T^{2} \) |
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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
−9.526338434163671432287910049454, −8.449361938606704849518686335332, −7.85713223500916950544549375410, −7.10880167259342825029625454000, −6.12301376777661502796714586192, −5.15807653924357763755017287994, −4.59005967740016512711309441741, −3.62009986146080255399130295807, −2.45120666802325773622392112246, −1.22936259812997770880375968335,
0.53218892069617798438680302840, 1.98441870594780553989992625647, 3.10240252970006382363844646452, 4.03044102888992278673656498503, 4.91255397785963325440714189745, 5.91762640976662757672093477668, 6.92540801797476806388187431463, 7.13253783689496077399818968359, 8.202257070419354569827824340390, 9.077819946483572983668023381304