L(s) = 1 | − 3i·3-s + (11.1 − 0.178i)5-s + 33.0i·7-s − 9·9-s + 48.3·11-s − 60.3i·13-s + (−0.536 − 33.5i)15-s + 17.7i·17-s + 130.·19-s + 99.2·21-s + 70.8i·23-s + (124. − 4i)25-s + 27i·27-s − 104.·29-s − 210.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.999 − 0.0160i)5-s + 1.78i·7-s − 0.333·9-s + 1.32·11-s − 1.28i·13-s + (−0.00923 − 0.577i)15-s + 0.253i·17-s + 1.58·19-s + 1.03·21-s + 0.642i·23-s + (0.999 − 0.0320i)25-s + 0.192i·27-s − 0.669·29-s − 1.21·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0160i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.97115 + 0.0157722i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97115 + 0.0157722i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 + (-11.1 + 0.178i)T \) |
good | 7 | \( 1 - 33.0iT - 343T^{2} \) |
| 11 | \( 1 - 48.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 17.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 70.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 104.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 210.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 300. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 240.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 108iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 278. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 328. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 889.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 241.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 103. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 277.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 274. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 366.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 57.7iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 203.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.28e3iT - 9.12e5T^{2} \) |
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
−12.81681192304772813016565827899, −12.20398311071718538353629575128, −11.10597781208159789355578545281, −9.459691099489557829666822620245, −8.999138419914933678607168032711, −7.52160920096974116156534185047, −5.94932090656668889734369235222, −5.53345386431883575173458458270, −3.01376935274297240641932811492, −1.59749681186365131846113244135,
1.34871222697443959106075745453, 3.62980877620348759504541606460, 4.74550838166045550077459762798, 6.39164619785762993858077139790, 7.31455869948494938565759026149, 9.164387088752878325423512724974, 9.740022623030560319470314271253, 10.80316602975846990912315281459, 11.75283543579525608574561280666, 13.35889354695434387026004174657