L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (−0.499 + 0.866i)12-s + 2·13-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + (0.499 + 0.866i)18-s + (−4 + 6.92i)19-s + 0.999·20-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (−0.144 + 0.249i)12-s + 0.554·13-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + (0.117 + 0.204i)18-s + (−0.917 + 1.58i)19-s + 0.223·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.540613602\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.540613602\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.791434880902513186132208580318, −8.430246770546019188023280837039, −8.121710921245702784045025082274, −6.87063015451002989658246319533, −6.13443791549414235761296963655, −5.46613137025875440917645636038, −4.16254106534565479744723962532, −3.50787849130044246930199504118, −2.25599956164279389955355448938, −1.19960498465657348157022178252,
0.65743226668061380058924481398, 2.67786113457138663816340637913, 3.71077312113999497484978498745, 4.78477328747497908082097298498, 5.09490089267121241179895457275, 6.33430221595517317096181717659, 6.86540020921682057705218605043, 8.004426382159837622076155639792, 8.614272226404186525909055427698, 9.449215582184132038465277622427