L(s) = 1 | + (1.06 − 0.926i)2-s + (0.284 − 1.97i)4-s + 2.85·5-s + i·7-s + (−1.52 − 2.37i)8-s + (3.05 − 2.64i)10-s + 4.98i·11-s + 2.35i·13-s + (0.926 + 1.06i)14-s + (−3.83 − 1.12i)16-s − 8.19i·17-s + 5.90·19-s + (0.813 − 5.65i)20-s + (4.61 + 5.32i)22-s + 4.43·23-s + ⋯ |
L(s) = 1 | + (0.755 − 0.654i)2-s + (0.142 − 0.989i)4-s + 1.27·5-s + 0.377i·7-s + (−0.540 − 0.841i)8-s + (0.966 − 0.837i)10-s + 1.50i·11-s + 0.652i·13-s + (0.247 + 0.285i)14-s + (−0.959 − 0.281i)16-s − 1.98i·17-s + 1.35·19-s + (0.181 − 1.26i)20-s + (0.984 + 1.13i)22-s + 0.925·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.326931145\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.326931145\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.06 + 0.926i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 2.85T + 5T^{2} \) |
| 11 | \( 1 - 4.98iT - 11T^{2} \) |
| 13 | \( 1 - 2.35iT - 13T^{2} \) |
| 17 | \( 1 + 8.19iT - 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 - 6.41T + 29T^{2} \) |
| 31 | \( 1 + 3.31iT - 31T^{2} \) |
| 37 | \( 1 + 5.51iT - 37T^{2} \) |
| 41 | \( 1 - 3.41iT - 41T^{2} \) |
| 43 | \( 1 - 1.37T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 10.6iT - 59T^{2} \) |
| 61 | \( 1 + 1.41iT - 61T^{2} \) |
| 67 | \( 1 - 0.221T + 67T^{2} \) |
| 71 | \( 1 + 0.398T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 9.76iT - 79T^{2} \) |
| 83 | \( 1 + 2.06iT - 83T^{2} \) |
| 89 | \( 1 - 17.6iT - 89T^{2} \) |
| 97 | \( 1 + 3.62T + 97T^{2} \) |
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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
−9.497579359702637776955726551895, −9.165371006241543549702218748858, −7.39326944272496285160303280186, −6.80660794019320379016254295426, −5.83237600176372002043594856210, −5.03879945028469794064698063228, −4.53340472373168207957872996657, −2.96069892812514495890978048655, −2.32942376430752696959926470730, −1.27575501263964985491544603786,
1.36024638331037826456394177186, 2.90383251480823021871740385854, 3.52459311206368728110847747547, 4.83346049332819098573530899633, 5.68875652510476740883693752182, 6.10518657742167293689098599981, 6.91246965357239653025162282890, 8.078562088870693862707251776570, 8.531553107147852331238137613853, 9.520159799445252846693592707669