L(s) = 1 | + (1.05 − 0.946i)2-s + (0.104 − 0.994i)4-s + (0.669 − 0.743i)5-s + (−0.575 − 1.29i)7-s − 1.41i·10-s + (−1.82 − 0.813i)14-s + (0.978 + 0.207i)16-s + (−0.669 − 0.743i)20-s + (−1.34 + 0.437i)28-s + (−0.575 − 1.29i)29-s + (−0.978 + 0.207i)31-s + (1.22 − 0.707i)32-s + (−1.34 − 0.437i)35-s + (0.809 + 0.587i)37-s + (−0.104 − 0.994i)47-s + ⋯ |
L(s) = 1 | + (1.05 − 0.946i)2-s + (0.104 − 0.994i)4-s + (0.669 − 0.743i)5-s + (−0.575 − 1.29i)7-s − 1.41i·10-s + (−1.82 − 0.813i)14-s + (0.978 + 0.207i)16-s + (−0.669 − 0.743i)20-s + (−1.34 + 0.437i)28-s + (−0.575 − 1.29i)29-s + (−0.978 + 0.207i)31-s + (1.22 − 0.707i)32-s + (−1.34 − 0.437i)35-s + (0.809 + 0.587i)37-s + (−0.104 − 0.994i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.343916802\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.343916802\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.05 + 0.946i)T + (0.104 - 0.994i)T^{2} \) |
| 5 | \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
| 7 | \( 1 + (0.575 + 1.29i)T + (-0.669 + 0.743i)T^{2} \) |
| 13 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.575 + 1.29i)T + (-0.669 + 0.743i)T^{2} \) |
| 31 | \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 61 | \( 1 + (0.294 - 1.38i)T + (-0.913 - 0.406i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 83 | \( 1 + (-0.294 + 1.38i)T + (-0.913 - 0.406i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
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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
−8.680067427408847053172905211245, −7.74797027178745503124232822515, −6.98176900749043630308108296630, −5.96001902948926300708780915407, −5.36828515649304079305583741161, −4.43410872995682391372424249113, −3.94042534878805396704697943075, −3.07184906043540010636249121471, −2.01891408509430050084110518319, −1.04670719947890724920250971628,
1.94081623471609119191244817187, 2.91902211633691124902515583535, 3.61079771852409650686819742790, 4.76553754087395162098475428010, 5.52388638661100122303524736811, 6.03909827845146663324927090882, 6.56578230249700112273726836755, 7.28773209552757397331998585155, 8.123401508322644585953541218371, 9.174060517176284005724373585430