L(s) = 1 | + (0.173 + 0.984i)3-s + (0.173 − 0.300i)7-s + (−0.939 + 0.342i)9-s + (−0.266 + 1.50i)13-s + (−0.5 + 0.866i)19-s + (0.326 + 0.118i)21-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 1.62i)31-s + 1.87·37-s − 1.53·39-s + (−1.43 + 1.20i)43-s + (0.439 + 0.761i)49-s + (−0.939 − 0.342i)57-s + (−1.17 − 0.984i)61-s + ⋯ |
L(s) = 1 | + (0.173 + 0.984i)3-s + (0.173 − 0.300i)7-s + (−0.939 + 0.342i)9-s + (−0.266 + 1.50i)13-s + (−0.5 + 0.866i)19-s + (0.326 + 0.118i)21-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 1.62i)31-s + 1.87·37-s − 1.53·39-s + (−1.43 + 1.20i)43-s + (0.439 + 0.761i)49-s + (−0.939 − 0.342i)57-s + (−1.17 − 0.984i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.077564551\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.077564551\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 1.87T + T^{2} \) |
| 41 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
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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
−9.053825368179503620994259110924, −8.361376034804593202694771043866, −7.65271504012113139768529705218, −6.64030168448643131010264260116, −6.03528575225854966802204188240, −4.92734291840790024920662330494, −4.42448343717046366068550183897, −3.71236625183290344143976766058, −2.71179615606206542573867004447, −1.64901373199787321529360415803,
0.58985391935057391222328414828, 1.94047746366871308628067987901, 2.73702328185048320712100368615, 3.55761877557224468747289257503, 4.80719214729394950553267013823, 5.64478416268704995682875999562, 6.14961110583822505872615969516, 7.21607407207777045919960294633, 7.61227897635776141824831585926, 8.368953822923575840755203396329