L(s) = 1 | − 0.732i·3-s − 2.73·7-s + 2.46·9-s + 2i·11-s − 3.46i·13-s + 3.46·17-s − 7.46i·19-s + 2i·21-s + 4.19·23-s − 4i·27-s − 6.92i·29-s − 1.46·31-s + 1.46·33-s + 2i·37-s − 2.53·39-s + ⋯ |
L(s) = 1 | − 0.422i·3-s − 1.03·7-s + 0.821·9-s + 0.603i·11-s − 0.960i·13-s + 0.840·17-s − 1.71i·19-s + 0.436i·21-s + 0.874·23-s − 0.769i·27-s − 1.28i·29-s − 0.262·31-s + 0.254·33-s + 0.328i·37-s − 0.406·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08374 - 0.831587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08374 - 0.831587i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 0.732iT - 3T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 7.46iT - 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + 8.73iT - 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 + 4.53iT - 53T^{2} \) |
| 59 | \( 1 - 0.535iT - 59T^{2} \) |
| 61 | \( 1 - 4.92iT - 61T^{2} \) |
| 67 | \( 1 - 7.26iT - 67T^{2} \) |
| 71 | \( 1 - 1.46T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 4.73iT - 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−9.977327795166027493730404044021, −9.440954098168652739442408428681, −8.332874592748855555366033911579, −7.24268959836692863489301134774, −6.87538798215349305034241455904, −5.73713869313813131291205159471, −4.69914098317013200964109716396, −3.48443302444588795703526055651, −2.42079350331799264682391429705, −0.74503790584754474829739254472,
1.46472811326261188731775748312, 3.21917270886900169233480065245, 3.87471415567531540069732163171, 5.07441855356215912810014916301, 6.12047737831562567021336883993, 6.91084938234565956411098373044, 7.85050516282872322483477902373, 8.999369819261444313779531801457, 9.617945962840799651094781508613, 10.32805371524719015848111463676