L(s) = 1 | + 1.58i·5-s + 5.24·7-s − 5.82i·11-s + 2.48·25-s + 2.82i·29-s − 0.757·31-s + 8.31i·35-s + 20.4·49-s + 10.0i·53-s + 9.24·55-s − 11.3i·59-s + 1.48·73-s − 30.5i·77-s + 10·79-s + 12.1i·83-s + ⋯ |
L(s) = 1 | + 0.709i·5-s + 1.98·7-s − 1.75i·11-s + 0.497·25-s + 0.525i·29-s − 0.136·31-s + 1.40i·35-s + 2.92·49-s + 1.38i·53-s + 1.24·55-s − 1.47i·59-s + 0.173·73-s − 3.48i·77-s + 1.12·79-s + 1.33i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95146\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95146\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.58iT - 5T^{2} \) |
| 7 | \( 1 - 5.24T + 7T^{2} \) |
| 11 | \( 1 + 5.82iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 0.757T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 1.48T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−10.56651078944524923318323683910, −9.132662705258513185782304852625, −8.349846044881212348279611486895, −7.80807682748925603889408302124, −6.75413502859813980266114784138, −5.69953147403795672436086624383, −4.92901224701463792521965998459, −3.74469558901146948320935472133, −2.60062037510907051134444169126, −1.21941888614437271042163266674,
1.36934781319433287747089471348, 2.24521132614957376419970949461, 4.20998673994309506133722944488, 4.77181274146763543101832909328, 5.45383661855659355418933396117, 6.94679222661206738265920979859, 7.72865430939787660855344514125, 8.400752174608832916405868815834, 9.242290689300269651250819024165, 10.18067502461990301745607553271