| L(s) = 1 | + i·3-s + (−2 − i)5-s − 4i·7-s − 9-s − 6i·13-s + (1 − 2i)15-s + 2i·17-s − 6·19-s + 4·21-s + 6i·23-s + (3 + 4i)25-s − i·27-s − 8·29-s + 8·31-s + (−4 + 8i)35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.894 − 0.447i)5-s − 1.51i·7-s − 0.333·9-s − 1.66i·13-s + (0.258 − 0.516i)15-s + 0.485i·17-s − 1.37·19-s + 0.872·21-s + 1.25i·23-s + (0.600 + 0.800i)25-s − 0.192i·27-s − 1.48·29-s + 1.43·31-s + (−0.676 + 1.35i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2 + i)T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
| 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
−8.543126658701964104958295717709, −8.010699152436124105473554624691, −7.41241904942930849113231665138, −6.39758296404270329271114699229, −5.32700806985746873764778142788, −4.48918154814559053980498185657, −3.81359982910875464120650501338, −3.11867098068120154056764373794, −1.23597930730920045812527062619, 0,
2.02776853267844431333687807267, 2.59113681909505243051019872563, 3.89038053893457419609344762087, 4.71854637639482915853153214759, 5.85749252372847187328194395210, 6.61225372139382135608161051678, 7.14324465888370392805523757025, 8.263606430521333597264332343543, 8.709637356074040664941971008270
