L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − i·8-s − 9-s + 4·11-s − i·12-s + 2i·13-s + 16-s + i·17-s − i·18-s − 4·19-s + 4i·22-s + 24-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + 1.20·11-s − 0.288i·12-s + 0.554i·13-s + 0.250·16-s + 0.242i·17-s − 0.235i·18-s − 0.917·19-s + 0.852i·22-s + 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.491646708\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.491646708\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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
−8.995115335819327596569615436316, −8.613504799018316812866139079237, −7.75770703475408108362581473403, −6.57062747236707499110170805583, −6.44662275511855786642609782790, −5.32752237855133606322428625067, −4.38015416747897394425645690374, −3.97416455850201745361862764946, −2.74497980197667083150885584168, −1.27540490215544479470173831041,
0.54509965950568993015144627592, 1.65516391802075173741387878117, 2.61511717379824294063797066338, 3.61090425059723521153176237571, 4.42594261847773308528203492736, 5.42294632743419596965832574709, 6.34909153615327659351796855786, 6.95805976449372361837424040819, 7.989757574387365598442964700160, 8.634357428742950836229566907781