| L(s) = 1 | − i·2-s − 4-s − 5-s + i·8-s + i·10-s + 5.37i·11-s + 1.19i·13-s + 16-s + 2.17·17-s + 2.71i·19-s + 20-s + 5.37·22-s − 0.496i·23-s + 25-s + 1.19·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447·5-s + 0.353i·8-s + 0.316i·10-s + 1.62i·11-s + 0.332i·13-s + 0.250·16-s + 0.528·17-s + 0.622i·19-s + 0.223·20-s + 1.14·22-s − 0.103i·23-s + 0.200·25-s + 0.235·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4862180117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4862180117\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 5.37iT - 11T^{2} \) |
| 13 | \( 1 - 1.19iT - 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 - 2.71iT - 19T^{2} \) |
| 23 | \( 1 + 0.496iT - 23T^{2} \) |
| 29 | \( 1 + 7.12iT - 29T^{2} \) |
| 31 | \( 1 - 0.550iT - 31T^{2} \) |
| 37 | \( 1 + 4.10T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 + 7.42T + 47T^{2} \) |
| 53 | \( 1 - 5.13iT - 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 1.88iT - 61T^{2} \) |
| 67 | \( 1 - 4.25T + 67T^{2} \) |
| 71 | \( 1 - 10.8iT - 71T^{2} \) |
| 73 | \( 1 + 15.5iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 7.65iT - 97T^{2} \) |
| 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
−8.671314733120965177489847240009, −7.83014581686521443507052267874, −7.35325085162106782777875299612, −6.44388041141618296393094717550, −5.51698926885002321976205143817, −4.57523881300526578804621365658, −4.15521655769665154025590408947, −3.18889660475540939126862107791, −2.21169160421251569535158879756, −1.38474003794844124781896300045,
0.14662132357616052823711778366, 1.26913024997026020767037833565, 2.95859239362047365341560993849, 3.46490600683587636078862423152, 4.45436089602621753078823337264, 5.36061824151873656914786938607, 5.84102240789288806955852039643, 6.75073813070082425687244843034, 7.33944412826290628019255218303, 8.244770145912321876235446614679
