| L(s) = 1 | + 9i·3-s − 6i·5-s + 40·7-s − 81·9-s + 564i·11-s + 638i·13-s + 54·15-s + 882·17-s − 556i·19-s + 360i·21-s + 840·23-s + 3.08e3·25-s − 729i·27-s + 4.63e3i·29-s + 4.40e3·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.107i·5-s + 0.308·7-s − 0.333·9-s + 1.40i·11-s + 1.04i·13-s + 0.0619·15-s + 0.740·17-s − 0.353i·19-s + 0.178i·21-s + 0.331·23-s + 0.988·25-s − 0.192i·27-s + 1.02i·29-s + 0.822·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.004786228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.004786228\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9iT \) |
| good | 5 | \( 1 + 6iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 40T + 1.68e4T^{2} \) |
| 11 | \( 1 - 564iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 638iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 882T + 1.41e6T^{2} \) |
| 19 | \( 1 + 556iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 840T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.63e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.40e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.41e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 6.87e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.64e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.86e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.37e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.80e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.97e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.30e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.24e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.94e4T + 8.58e9T^{2} \) |
| 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
−9.823437654447782065812759152110, −9.173170386149063368987707856146, −8.291899371157556342161880658063, −7.22274966924642129250800959790, −6.51953014637753083208115118758, −5.08281673098641020054229883864, −4.67387041217479497818589474547, −3.56137268484040861721727571481, −2.32104286521982187810469123654, −1.20606588853701860050365350773,
0.45450408718379859424597250660, 1.24658046380586815782549472029, 2.72741667749147320916382124252, 3.44155989069065292956407713337, 4.89797196547844109026424372347, 5.84153358745822917922678003484, 6.48859941483514712258858333953, 7.84875513345970824594419891266, 8.077229655876704661869720726343, 9.130075978481447225822245940378
