L(s) = 1 | + 4.47i·3-s + 31.3i·7-s + 6.99·9-s − 8.94·11-s − 62i·13-s + 46i·17-s − 107.·19-s − 140·21-s + 192. i·23-s + 152. i·27-s + 90·29-s − 152.·31-s − 40.0i·33-s + 214i·37-s + 277.·39-s + ⋯ |
L(s) = 1 | + 0.860i·3-s + 1.69i·7-s + 0.259·9-s − 0.245·11-s − 1.32i·13-s + 0.656i·17-s − 1.29·19-s − 1.45·21-s + 1.74i·23-s + 1.08i·27-s + 0.576·29-s − 0.880·31-s − 0.211i·33-s + 0.950i·37-s + 1.13·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9737835207\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9737835207\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.47iT - 27T^{2} \) |
| 7 | \( 1 - 31.3iT - 343T^{2} \) |
| 11 | \( 1 + 8.94T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 46iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 214iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 10T + 6.89e4T^{2} \) |
| 43 | \( 1 + 67.0iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 398. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 678iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 250T + 2.26e5T^{2} \) |
| 67 | \( 1 + 49.1iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 366.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 522iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 876.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 380. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 970T + 7.04e5T^{2} \) |
| 97 | \( 1 - 934iT - 9.12e5T^{2} \) |
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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
−10.21648314435144520079304243570, −9.630085796624385972814856461537, −8.644854675887144877967830322096, −8.149248720259187538052384987214, −6.81105134158700328958139090075, −5.57544084503013976553131607298, −5.28866899065239042374742781497, −3.95403084669979394036208221374, −2.98117426256350512159249578057, −1.80087276109114388990115333621,
0.25815319372157042234637789427, 1.33194586535707554332950589761, 2.46713464785166697088206785957, 4.14150390545467274529921612365, 4.50594532920795169653809638031, 6.24934413170939122678204713825, 6.92841329893006576580853625040, 7.40568510119078044886644859483, 8.380547769992236014226950755942, 9.394392308519436368264363520578