L(s) = 1 | − 1.21i·5-s + (−0.355 + 2.62i)7-s − 4.06·11-s − 1.42·13-s − 4.74·17-s + 4.84·19-s + 6.03i·23-s + 3.51·25-s − 3.45·29-s − 3.15i·31-s + (3.19 + 0.432i)35-s − 9.24i·37-s − 1.24·41-s − 3.23i·43-s + 9.43·47-s + ⋯ |
L(s) = 1 | − 0.544i·5-s + (−0.134 + 0.990i)7-s − 1.22·11-s − 0.396·13-s − 1.15·17-s + 1.11·19-s + 1.25i·23-s + 0.703·25-s − 0.641·29-s − 0.566i·31-s + (0.539 + 0.0731i)35-s − 1.51i·37-s − 0.194·41-s − 0.492i·43-s + 1.37·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.058995002\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.058995002\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.355 - 2.62i)T \) |
good | 5 | \( 1 + 1.21iT - 5T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 + 1.42T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 4.84T + 19T^{2} \) |
| 23 | \( 1 - 6.03iT - 23T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 + 3.15iT - 31T^{2} \) |
| 37 | \( 1 + 9.24iT - 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 3.23iT - 43T^{2} \) |
| 47 | \( 1 - 9.43T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 + 10.2iT - 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 6.51iT - 67T^{2} \) |
| 71 | \( 1 + 2.14iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 16.3iT - 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 11.4iT - 97T^{2} \) |
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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
−8.311844843270574886968370634471, −7.58586182024005138801388786914, −6.95577743041931419337555513212, −5.73490164073761660694220954440, −5.42606434046288158138375404083, −4.69533488734569113559356156656, −3.59650304984038783197986622526, −2.62551161065801344378300504131, −1.91758724731194960418676926972, −0.35355527666997524368993295093,
0.943392275461088432903916936636, 2.42844595781995552596693600798, 3.00484675454924776860435688818, 4.06526718334875782509968855225, 4.82450592729185736921198915859, 5.55184677908795590697304747732, 6.80127696444119704267215595274, 6.90895854238165573010259126855, 7.83679207942340779744869522118, 8.449206687712883373985654287656