L(s) = 1 | + (−0.0660 + 1.41i)2-s + 0.496i·3-s + (−1.99 − 0.186i)4-s + (−0.987 + 2.00i)5-s + (−0.701 − 0.0328i)6-s + (1.55 + 1.55i)7-s + (0.395 − 2.80i)8-s + 2.75·9-s + (−2.76 − 1.52i)10-s + (−4.19 − 4.19i)11-s + (0.0927 − 0.988i)12-s + 5.09·13-s + (−2.29 + 2.09i)14-s + (−0.996 − 0.490i)15-s + (3.93 + 0.743i)16-s + (0.213 + 0.213i)17-s + ⋯ |
L(s) = 1 | + (−0.0467 + 0.998i)2-s + 0.286i·3-s + (−0.995 − 0.0933i)4-s + (−0.441 + 0.897i)5-s + (−0.286 − 0.0133i)6-s + (0.587 + 0.587i)7-s + (0.139 − 0.990i)8-s + 0.917·9-s + (−0.875 − 0.482i)10-s + (−1.26 − 1.26i)11-s + (0.0267 − 0.285i)12-s + 1.41·13-s + (−0.614 + 0.559i)14-s + (−0.257 − 0.126i)15-s + (0.982 + 0.185i)16-s + (0.0517 + 0.0517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.510684 + 0.693990i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.510684 + 0.693990i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0660 - 1.41i)T \) |
| 5 | \( 1 + (0.987 - 2.00i)T \) |
good | 3 | \( 1 - 0.496iT - 3T^{2} \) |
| 7 | \( 1 + (-1.55 - 1.55i)T + 7iT^{2} \) |
| 11 | \( 1 + (4.19 + 4.19i)T + 11iT^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + (-0.213 - 0.213i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.844 + 0.844i)T + 19iT^{2} \) |
| 23 | \( 1 + (-1.70 + 1.70i)T - 23iT^{2} \) |
| 29 | \( 1 + (2.24 - 2.24i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.818iT - 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 3.34iT - 41T^{2} \) |
| 43 | \( 1 + 4.49T + 43T^{2} \) |
| 47 | \( 1 + (4.29 - 4.29i)T - 47iT^{2} \) |
| 53 | \( 1 - 1.00iT - 53T^{2} \) |
| 59 | \( 1 + (-7.65 + 7.65i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.90 + 1.90i)T + 61iT^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + (2.70 + 2.70i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 - 9.17iT - 83T^{2} \) |
| 89 | \( 1 - 4.25T + 89T^{2} \) |
| 97 | \( 1 + (7.15 + 7.15i)T + 97iT^{2} \) |
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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
−15.00254247836111967447988430152, −13.86141517107041202774782899851, −12.92884301188020011244122391704, −11.17907618318975396542052236527, −10.36108516246407036592889439226, −8.720407871688205963211510433731, −7.88343028593941408125589062355, −6.53539307115159497819393723687, −5.26148760688914238905177044081, −3.59452952640116843501973373663,
1.54532472257734829377951785688, 4.02379646555656430446560006042, 5.06364954192457090886325553587, 7.49765615840283016594913313476, 8.422872581917627559009677318301, 9.844593377480769660702391727111, 10.81899674984462984771507647415, 11.99534869686843027643777202302, 13.03462503226843705109011184835, 13.42949227417018672847499534794