L(s) = 1 | + (−0.221 − 0.827i)3-s + (−1.60 − 1.55i)5-s + (1.42 − 2.22i)7-s + (1.96 − 1.13i)9-s + (1.59 − 2.75i)11-s + (−4.30 + 4.30i)13-s + (−0.930 + 1.67i)15-s + (0.267 − 0.0717i)17-s + (2.95 + 5.11i)19-s + (−2.16 − 0.686i)21-s + (1.51 − 5.64i)23-s + (0.163 + 4.99i)25-s + (−3.19 − 3.19i)27-s + 9.49i·29-s + (3.16 + 1.82i)31-s + ⋯ |
L(s) = 1 | + (−0.128 − 0.477i)3-s + (−0.718 − 0.695i)5-s + (0.539 − 0.842i)7-s + (0.654 − 0.377i)9-s + (0.479 − 0.830i)11-s + (−1.19 + 1.19i)13-s + (−0.240 + 0.432i)15-s + (0.0649 − 0.0173i)17-s + (0.677 + 1.17i)19-s + (−0.471 − 0.149i)21-s + (0.315 − 1.17i)23-s + (0.0327 + 0.999i)25-s + (−0.613 − 0.613i)27-s + 1.76i·29-s + (0.567 + 0.327i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.819773 - 0.586266i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.819773 - 0.586266i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.60 + 1.55i)T \) |
| 7 | \( 1 + (-1.42 + 2.22i)T \) |
good | 3 | \( 1 + (0.221 + 0.827i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-1.59 + 2.75i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.30 - 4.30i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.267 + 0.0717i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.95 - 5.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.51 + 5.64i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 9.49iT - 29T^{2} \) |
| 31 | \( 1 + (-3.16 - 1.82i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.653 + 0.175i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 1.09iT - 41T^{2} \) |
| 43 | \( 1 + (-4.70 - 4.70i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.118 - 0.443i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.96 - 0.526i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-4.29 + 7.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.23 + 4.75i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0180 - 0.0674i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 5.14T + 71T^{2} \) |
| 73 | \( 1 + (-0.446 - 1.66i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (5.31 - 3.06i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.30 - 1.30i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.19 - 2.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.65 - 5.65i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.70975450060064940825740105366, −12.08121111088494129571132012946, −11.16112157988023460988015320309, −9.834190369704800375607060265143, −8.645827958273667584693493770025, −7.54028359628198997774656859721, −6.70506579902386931212380830710, −4.89440575677926695590866883602, −3.84382899481305712348472687809, −1.23815652034621334078932202121,
2.62168941232299601794351779750, 4.33274928660251847485286996969, 5.39868007962468043690575674805, 7.16812269098289044909251602063, 7.86317285141959704176227673507, 9.444371283219081786415363983895, 10.25459332821783486960144946904, 11.44587339697207163735392897830, 12.08451749588257973845916081892, 13.30057594362835121083370646262