L(s) = 1 | − i·7-s − 4.24i·11-s − 6i·13-s − 4.24·23-s − 5·25-s − 4.24·29-s − 4i·31-s + 6i·37-s + 8.48i·41-s − 6·43-s − 8.48·47-s − 49-s + 12.7·53-s + 8.48i·59-s + 6i·61-s + ⋯ |
L(s) = 1 | − 0.377i·7-s − 1.27i·11-s − 1.66i·13-s − 0.884·23-s − 25-s − 0.787·29-s − 0.718i·31-s + 0.986i·37-s + 1.32i·41-s − 0.914·43-s − 1.23·47-s − 0.142·49-s + 1.74·53-s + 1.10i·59-s + 0.768i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5046993528\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5046993528\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 8.48iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 - 16.9iT - 83T^{2} \) |
| 89 | \( 1 - 8.48iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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
−8.140630737945495921050259538070, −7.50768685595505047311050377660, −6.46384623162255478689977856472, −5.79252410295276734172617003258, −5.25287280722824407309816137669, −4.08020597178771150561004771456, −3.40449678285312499520422442216, −2.59418935690332892199898002178, −1.23445644086025279572211032659, −0.14228804874971602617736121744,
1.86315077615667525680878371579, 2.10110872134390969165841895213, 3.62116990776835524145782518484, 4.24225296431010947518190985975, 5.06279258406415346869821531979, 5.87472715297778390839506675012, 6.80142377286183488941458998056, 7.20020020880583669837973944067, 8.121038423024082749760834662193, 8.882573728797241316488891784238