L(s) = 1 | + (1.41 + i)3-s + (−1 + 2i)5-s + (0.414 − 0.414i)7-s + (1.00 + 2.82i)9-s − 4.82i·11-s + (−1.82 − 1.82i)13-s + (−3.41 + 1.82i)15-s + (3.82 + 3.82i)17-s − 4.82i·19-s + (1 − 0.171i)21-s + (1.58 − 1.58i)23-s + (−3 − 4i)25-s + (−1.41 + 5.00i)27-s − 7.65·29-s − 5.65·31-s + ⋯ |
L(s) = 1 | + (0.816 + 0.577i)3-s + (−0.447 + 0.894i)5-s + (0.156 − 0.156i)7-s + (0.333 + 0.942i)9-s − 1.45i·11-s + (−0.507 − 0.507i)13-s + (−0.881 + 0.472i)15-s + (0.928 + 0.928i)17-s − 1.10i·19-s + (0.218 − 0.0374i)21-s + (0.330 − 0.330i)23-s + (−0.600 − 0.800i)25-s + (−0.272 + 0.962i)27-s − 1.42·29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17079 + 0.443025i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17079 + 0.443025i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| 5 | \( 1 + (1 - 2i)T \) |
good | 7 | \( 1 + (-0.414 + 0.414i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.82iT - 11T^{2} \) |
| 13 | \( 1 + (1.82 + 1.82i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.82 - 3.82i)T + 17iT^{2} \) |
| 19 | \( 1 + 4.82iT - 19T^{2} \) |
| 23 | \( 1 + (-1.58 + 1.58i)T - 23iT^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + (0.171 - 0.171i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-2.41 - 2.41i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.41 - 6.41i)T + 47iT^{2} \) |
| 53 | \( 1 + (3 - 3i)T - 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + (4.07 - 4.07i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.48iT - 71T^{2} \) |
| 73 | \( 1 + (-6.65 - 6.65i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.82iT - 79T^{2} \) |
| 83 | \( 1 + (5.24 - 5.24i)T - 83iT^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 + (-1 + i)T - 97iT^{2} \) |
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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
−13.88345756400012474033439340038, −12.73141688468361656524898988230, −11.10075595154082347522525958420, −10.67129694401731009154172331954, −9.346263315336985112520125717747, −8.200516864596251416301285236798, −7.31485467661676591988882119203, −5.63810030527158129573672484423, −3.89332655317309448156251699698, −2.85965166053531634219261317474,
1.84911852831014022172884294492, 3.83608124445814178280715476541, 5.25311838401986642963567547365, 7.18618872469779098628965231316, 7.82123796283941717276572096540, 9.130846668355472812328543584266, 9.806461743169845918341700428043, 11.77417577058002925307220746110, 12.37694875376035493538967569664, 13.21555500729749072517336610370