L(s) = 1 | + 4.04·5-s − 3.89i·7-s + 0.468i·11-s + 4.89i·13-s + 3.57i·17-s − 1.89·19-s + 8.56·23-s + 11.3·25-s − 2.36·29-s + 7.18i·31-s − 15.7i·35-s + 6.16i·37-s + 5.13i·41-s − 10.5·43-s + 9.37·47-s + ⋯ |
L(s) = 1 | + 1.80·5-s − 1.47i·7-s + 0.141i·11-s + 1.35i·13-s + 0.867i·17-s − 0.435·19-s + 1.78·23-s + 2.27·25-s − 0.439·29-s + 1.28i·31-s − 2.66i·35-s + 1.01i·37-s + 0.801i·41-s − 1.60·43-s + 1.36·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.970040693\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.970040693\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 4.04T + 5T^{2} \) |
| 7 | \( 1 + 3.89iT - 7T^{2} \) |
| 11 | \( 1 - 0.468iT - 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 - 3.57iT - 17T^{2} \) |
| 19 | \( 1 + 1.89T + 19T^{2} \) |
| 23 | \( 1 - 8.56T + 23T^{2} \) |
| 29 | \( 1 + 2.36T + 29T^{2} \) |
| 31 | \( 1 - 7.18iT - 31T^{2} \) |
| 37 | \( 1 - 6.16iT - 37T^{2} \) |
| 41 | \( 1 - 5.13iT - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 9.37T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 - 6.26iT - 59T^{2} \) |
| 61 | \( 1 - 0.983iT - 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 3.02T + 71T^{2} \) |
| 73 | \( 1 - 4.58T + 73T^{2} \) |
| 79 | \( 1 + 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 5.66iT - 83T^{2} \) |
| 89 | \( 1 + 3.17iT - 89T^{2} \) |
| 97 | \( 1 - 0.611T + 97T^{2} \) |
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
−8.420794940147371442242443904125, −7.18958278918857126729056752292, −6.76740040731968435117562318982, −6.29790003961313329102275199847, −5.26026974390277613571658878256, −4.66749620149250115171710431897, −3.80440763380192704787253577336, −2.78092386312246623024769015200, −1.70281048922341112274259226157, −1.20404778432676945936821468410,
0.834227260030314766597297739188, 2.19257093701751810019228929889, 2.49247258030340431590569405106, 3.38631198176706170706312390711, 4.94348296122514984160251444141, 5.46617078027230086147212725697, 5.75996032202090777014551354588, 6.58350726759647394407749058359, 7.37163130584375319950900301314, 8.476712901151108289424468190631