L(s) = 1 | + (−3.12 − 0.837i)3-s + (1.01 + 1.99i)5-s + (0.870 + 2.49i)7-s + (6.46 + 3.73i)9-s + (0.615 + 1.06i)11-s + (−2.44 + 2.44i)13-s + (−1.48 − 7.07i)15-s + (0.395 − 1.47i)17-s + (−2.14 + 3.71i)19-s + (−0.626 − 8.53i)21-s + (0.999 − 0.267i)23-s + (−2.95 + 4.03i)25-s + (−10.2 − 10.2i)27-s − 3.02i·29-s + (4.67 − 2.70i)31-s + ⋯ |
L(s) = 1 | + (−1.80 − 0.483i)3-s + (0.452 + 0.892i)5-s + (0.328 + 0.944i)7-s + (2.15 + 1.24i)9-s + (0.185 + 0.321i)11-s + (−0.677 + 0.677i)13-s + (−0.384 − 1.82i)15-s + (0.0960 − 0.358i)17-s + (−0.492 + 0.853i)19-s + (−0.136 − 1.86i)21-s + (0.208 − 0.0558i)23-s + (−0.591 + 0.806i)25-s + (−1.96 − 1.96i)27-s − 0.562i·29-s + (0.840 − 0.485i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.539992 + 0.358050i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.539992 + 0.358050i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.01 - 1.99i)T \) |
| 7 | \( 1 + (-0.870 - 2.49i)T \) |
good | 3 | \( 1 + (3.12 + 0.837i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-0.615 - 1.06i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.44 - 2.44i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.395 + 1.47i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (2.14 - 3.71i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.999 + 0.267i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 3.02iT - 29T^{2} \) |
| 31 | \( 1 + (-4.67 + 2.70i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.328 - 1.22i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 1.26iT - 41T^{2} \) |
| 43 | \( 1 + (-6.08 - 6.08i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.36 + 1.43i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.21 + 12.0i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (1.71 + 2.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.82 + 2.78i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.04 + 1.08i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + (-5.07 - 1.36i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.35 - 2.51i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.84 + 8.84i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.05 + 5.29i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.1 - 11.1i)T + 97iT^{2} \) |
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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
−13.04502993715146928025293324311, −12.00968429167243450346729701548, −11.55731389436805773616767458462, −10.50801248338225534162783407020, −9.575893680526473810291875080435, −7.65431445297840182829213068480, −6.55557597169277550532020519806, −5.86101631008835839264147049627, −4.69368239203309818440174139807, −2.06635450120563584829682270020,
0.860537810009495429094368963438, 4.28807609684337446176690496324, 5.10857076975769081523781786454, 6.12483718364897075003343414168, 7.36405270115491398756017211830, 9.055290431908148190923377838065, 10.30405605696663182443789198232, 10.77028745733891122991440700086, 11.97458287508504939782684400161, 12.68703770473068183160120744460