L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.453 + 0.891i)5-s + (−0.587 + 0.809i)8-s + (−0.707 + 0.707i)9-s + (0.156 − 0.987i)10-s + (−0.355 + 0.303i)13-s + (0.309 − 0.951i)16-s + (−1.29 + 0.794i)17-s + (0.453 − 0.891i)18-s + (0.156 + 0.987i)20-s + (−0.587 − 0.809i)25-s + (0.243 − 0.398i)26-s + (0.178 + 0.744i)29-s + i·32-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.453 + 0.891i)5-s + (−0.587 + 0.809i)8-s + (−0.707 + 0.707i)9-s + (0.156 − 0.987i)10-s + (−0.355 + 0.303i)13-s + (0.309 − 0.951i)16-s + (−1.29 + 0.794i)17-s + (0.453 − 0.891i)18-s + (0.156 + 0.987i)20-s + (−0.587 − 0.809i)25-s + (0.243 − 0.398i)26-s + (0.178 + 0.744i)29-s + i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4027266970\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4027266970\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (0.453 - 0.891i)T \) |
| 41 | \( 1 + (-0.951 + 0.309i)T \) |
good | 3 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (0.987 - 0.156i)T^{2} \) |
| 11 | \( 1 + (-0.453 + 0.891i)T^{2} \) |
| 13 | \( 1 + (0.355 - 0.303i)T + (0.156 - 0.987i)T^{2} \) |
| 17 | \( 1 + (1.29 - 0.794i)T + (0.453 - 0.891i)T^{2} \) |
| 19 | \( 1 + (-0.987 + 0.156i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (-0.178 - 0.744i)T + (-0.891 + 0.453i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.183 - 1.16i)T + (-0.951 - 0.309i)T^{2} \) |
| 43 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 47 | \( 1 + (-0.987 - 0.156i)T^{2} \) |
| 53 | \( 1 + (0.0819 - 0.133i)T + (-0.453 - 0.891i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2} \) |
| 67 | \( 1 + (0.891 - 0.453i)T^{2} \) |
| 71 | \( 1 + (0.453 - 0.891i)T^{2} \) |
| 73 | \( 1 + 0.312T + T^{2} \) |
| 79 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (1.40 + 1.20i)T + (0.156 + 0.987i)T^{2} \) |
| 97 | \( 1 + (-1.65 + 0.398i)T + (0.891 - 0.453i)T^{2} \) |
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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
−10.71575663096101590012977431611, −10.00413352145978645837839345223, −8.870422418042184164442511411292, −8.271835496884370613049443740886, −7.38289312071140798477759221708, −6.65521737418858050582876598916, −5.81746083515091577478259866575, −4.54732721342845712843693906207, −3.02408178600221431256045396979, −2.04988732026400040769542014670,
0.52630451104177643059474481458, 2.26198746414356185445250173190, 3.47172129686705503810237623599, 4.61324391963298476366394941671, 5.86971913920598927665399182954, 6.88467274200435391719714217203, 7.81447093990793772339087620188, 8.596100703075328495838275688927, 9.217803458260765221855356852427, 9.860500103249012817403534933395