| L(s) = 1 | − 6.32i·3-s + 18.9i·7-s − 13.0·9-s − 12.6·11-s − 38i·13-s + 34i·17-s + 101.·19-s + 120.·21-s − 82.2i·23-s − 88.5i·27-s − 270·29-s − 341.·31-s + 80.0i·33-s + 206i·37-s − 240.·39-s + ⋯ |
| L(s) = 1 | − 1.21i·3-s + 1.02i·7-s − 0.481·9-s − 0.346·11-s − 0.810i·13-s + 0.485i·17-s + 1.22·19-s + 1.24·21-s − 0.745i·23-s − 0.631i·27-s − 1.72·29-s − 1.97·31-s + 0.422i·33-s + 0.915i·37-s − 0.986·39-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
| \(L(2)\) |
\(\approx\) |
\(0.4291564591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4291564591\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 6.32iT - 27T^{2} \) |
| 7 | \( 1 - 18.9iT - 343T^{2} \) |
| 11 | \( 1 + 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 34iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 82.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 270T + 2.43e4T^{2} \) |
| 31 | \( 1 + 341.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 206iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 270T + 6.89e4T^{2} \) |
| 43 | \( 1 + 537. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 132. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 258iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 75.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 250T + 2.26e5T^{2} \) |
| 67 | \( 1 + 815. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 645.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 278.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 890T + 7.04e5T^{2} \) |
| 97 | \( 1 + 254iT - 9.12e5T^{2} \) |
| 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
−9.220870065530176281662890971546, −8.401264875865648820925650856946, −7.60185388116605287069103313047, −6.92927615535242650374301986898, −5.76984612327820124771060718543, −5.31227506199860961893728034562, −3.60044272933833869922958362188, −2.45081964971306873015103317937, −1.53460518927197358325607773591, −0.11045903100793846961740967733,
1.57389152440716541784844281009, 3.31055592878097786517778576309, 3.96405184522719603769132011787, 4.89245526847840738337020848588, 5.70100986867898217778300924042, 7.19161223319466253776772254730, 7.57104369640302537098902149200, 9.113680868192422248219708784918, 9.490673167313879611282885663190, 10.28738407076092521770518812439
