L(s) = 1 | + (−0.951 − 0.309i)5-s + (−0.587 − 0.809i)9-s + (−0.278 + 1.76i)13-s + (0.142 + 0.278i)17-s + (0.809 + 0.587i)25-s + (0.5 + 1.53i)29-s + (0.309 + 0.0489i)37-s + (1.53 − 1.11i)41-s + (0.309 + 0.951i)45-s + i·49-s + (0.809 + 0.412i)53-s + (0.363 − 0.5i)61-s + (0.809 − 1.58i)65-s + (0.278 + 1.76i)73-s + (−0.309 + 0.951i)81-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)5-s + (−0.587 − 0.809i)9-s + (−0.278 + 1.76i)13-s + (0.142 + 0.278i)17-s + (0.809 + 0.587i)25-s + (0.5 + 1.53i)29-s + (0.309 + 0.0489i)37-s + (1.53 − 1.11i)41-s + (0.309 + 0.951i)45-s + i·49-s + (0.809 + 0.412i)53-s + (0.363 − 0.5i)61-s + (0.809 − 1.58i)65-s + (0.278 + 1.76i)73-s + (−0.309 + 0.951i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8611454596\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8611454596\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.951 + 0.309i)T \) |
good | 3 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.142 - 0.278i)T + (-0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \) |
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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
−9.038298705677380986965994016795, −8.309988287284790126484151002659, −7.36424784891371032826396932828, −6.82082594240711813703756384934, −5.99975562659297493485732872761, −4.99772626019275995714293186367, −4.18276700272241489947510521298, −3.58691417295652220641385373025, −2.49769338831887000587190540143, −1.13245883968908674776834178519,
0.61201028299603333833511605368, 2.46754587853950075136474218024, 3.03438608557176455827680543950, 4.04468324140963730380499653137, 4.93702545966572107763287778313, 5.64958604735750515237201100204, 6.50258271844231974747592392518, 7.63443651581686307659377437055, 7.85823705431046114347194033415, 8.468406908616442160590598974611