| L(s) = 1 | + 4.51i·3-s + 50.0·7-s + 60.5·9-s − 158.·11-s − 97.3·13-s − 285. i·17-s + 414.·19-s + 226. i·21-s − 546.·23-s + 639. i·27-s − 1.16e3i·29-s − 369. i·31-s − 717. i·33-s − 1.42e3·37-s − 439. i·39-s + ⋯ |
| L(s) = 1 | + 0.502i·3-s + 1.02·7-s + 0.747·9-s − 1.31·11-s − 0.575·13-s − 0.987i·17-s + 1.14·19-s + 0.513i·21-s − 1.03·23-s + 0.877i·27-s − 1.38i·29-s − 0.384i·31-s − 0.659i·33-s − 1.03·37-s − 0.289i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.073559434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.073559434\) |
| \(L(3)\) |
|
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 | \( 1 - 4.51iT - 81T^{2} \) |
| 7 | \( 1 - 50.0T + 2.40e3T^{2} \) |
| 11 | \( 1 + 158.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 97.3T + 2.85e4T^{2} \) |
| 17 | \( 1 + 285. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 414.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 546.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.16e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 369. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.42e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.90e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 905. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 3.02e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 1.60e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.80e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 1.11e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 3.91e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.22e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.66e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 6.81e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 3.16e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.07e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.00e4iT - 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.792178774196842762276395959239, −8.513351148543799289588739448176, −7.68122847046051658503390714262, −7.13425977524284615492806740972, −5.55377918599067146687433900157, −4.98401655014809427996923453441, −4.13623177884822519228932403586, −2.81612516517230272188179903067, −1.71907466850914227694306956022, −0.23304485324746792940006185561,
1.31701818057558217435292253035, 2.10635030105684518549194147346, 3.45481697960151547367368764415, 4.77414676907637923434852670119, 5.34817386145970927657252516251, 6.59415137360480149977711809689, 7.62988824828070185166440467417, 7.911340229185042509321855714186, 8.972516538610908828454755737322, 10.25223112476033840010487416643
