L(s) = 1 | + (0.987 − 0.156i)2-s + (−0.156 + 0.987i)3-s + (0.951 − 0.309i)4-s + i·6-s + (0.891 − 0.453i)8-s + (−0.951 − 0.309i)9-s + (0.119 + 1.51i)11-s + (0.156 + 0.987i)12-s + (0.809 − 0.587i)16-s + (−0.581 − 0.497i)17-s + (−0.987 − 0.156i)18-s + (0.453 − 0.108i)19-s + (0.355 + 1.47i)22-s + (0.309 + 0.951i)24-s + (−0.587 − 0.809i)25-s + ⋯ |
L(s) = 1 | + (0.987 − 0.156i)2-s + (−0.156 + 0.987i)3-s + (0.951 − 0.309i)4-s + i·6-s + (0.891 − 0.453i)8-s + (−0.951 − 0.309i)9-s + (0.119 + 1.51i)11-s + (0.156 + 0.987i)12-s + (0.809 − 0.587i)16-s + (−0.581 − 0.497i)17-s + (−0.987 − 0.156i)18-s + (0.453 − 0.108i)19-s + (0.355 + 1.47i)22-s + (0.309 + 0.951i)24-s + (−0.587 − 0.809i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.721721216\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.721721216\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.987 + 0.156i)T \) |
| 3 | \( 1 + (0.156 - 0.987i)T \) |
| 41 | \( 1 + (0.453 - 0.891i)T \) |
good | 5 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 7 | \( 1 + (0.453 + 0.891i)T^{2} \) |
| 11 | \( 1 + (-0.119 - 1.51i)T + (-0.987 + 0.156i)T^{2} \) |
| 13 | \( 1 + (-0.891 - 0.453i)T^{2} \) |
| 17 | \( 1 + (0.581 + 0.497i)T + (0.156 + 0.987i)T^{2} \) |
| 19 | \( 1 + (-0.453 + 0.108i)T + (0.891 - 0.453i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.156 + 0.987i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (0.183 + 1.16i)T + (-0.951 + 0.309i)T^{2} \) |
| 47 | \( 1 + (-0.453 + 0.891i)T^{2} \) |
| 53 | \( 1 + (0.156 - 0.987i)T^{2} \) |
| 59 | \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 67 | \( 1 + (0.0123 - 0.156i)T + (-0.987 - 0.156i)T^{2} \) |
| 71 | \( 1 + (-0.987 + 0.156i)T^{2} \) |
| 73 | \( 1 + (1.14 + 1.14i)T + iT^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 - 0.618iT - T^{2} \) |
| 89 | \( 1 + (1.70 + 1.04i)T + (0.453 + 0.891i)T^{2} \) |
| 97 | \( 1 + (-1.93 - 0.152i)T + (0.987 + 0.156i)T^{2} \) |
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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
−10.17970529328425764604394738239, −9.877812069436317940642453224277, −8.808890603983176494598163700380, −7.54934209530432898090432529569, −6.71597001867647654590692632051, −5.74973866685601761531676150647, −4.77294887847317314222049670890, −4.32811448299598671114393967143, −3.20270496577922008199926678388, −2.07857854555378119213110535092,
1.53213415896797882060669003342, 2.83850252366302666433846365852, 3.71491001170435826466634894043, 5.08712593964411875968847942137, 5.94176871262834282265315475881, 6.44939999498317204846818081298, 7.46983224178529248138498896229, 8.150441739503839107120256270736, 9.012682207755177686518957063945, 10.48169674389389132539541282616