L(s) = 1 | + (−1 − 2i)5-s + (1 − i)13-s + (−5 − 5i)17-s + (−3 + 4i)25-s − 10i·29-s + (−7 − 7i)37-s − 10·41-s + 7i·49-s + (5 − 5i)53-s + 12·61-s + (−3 − i)65-s + (11 − 11i)73-s + (−5 + 15i)85-s + 10i·89-s + (−13 − 13i)97-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.894i)5-s + (0.277 − 0.277i)13-s + (−1.21 − 1.21i)17-s + (−0.600 + 0.800i)25-s − 1.85i·29-s + (−1.15 − 1.15i)37-s − 1.56·41-s + i·49-s + (0.686 − 0.686i)53-s + 1.53·61-s + (−0.372 − 0.124i)65-s + (1.28 − 1.28i)73-s + (−0.542 + 1.62i)85-s + 1.05i·89-s + (−1.31 − 1.31i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.458065 - 0.821587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.458065 - 0.821587i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
good | 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 + (5 + 5i)T + 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 10iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (7 + 7i)T + 37iT^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-5 + 5i)T - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-11 + 11i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 + (13 + 13i)T + 97iT^{2} \) |
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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
−10.01494075156774709424674860623, −9.155922408831269568252927866396, −8.480562004187249511904055025426, −7.60198276477324176105330257627, −6.65049688413170539207160463909, −5.46976334895499297820517657150, −4.63988777221551882064743399918, −3.67102638766190293415429729947, −2.18720762862321371885776946629, −0.47540063476749427317373765702,
1.86923672087254418009846353794, 3.23630675826133321837138035102, 4.09557281354080656170450891442, 5.31758959749386313518572698943, 6.63553547403116105017187727342, 6.91800161053221953059438822094, 8.256940296972529450224820556715, 8.780335037974028390004830251118, 10.08844439675758383541538758720, 10.69070048906599955411700646339