L(s) = 1 | + (1.21 + 0.885i)2-s + (0.590 + 1.62i)3-s + (0.0828 + 0.254i)4-s + (1.71 + 2.35i)5-s + (−0.721 + 2.50i)6-s + (−2.68 + 0.874i)7-s + (0.806 − 2.48i)8-s + (−2.30 + 1.92i)9-s + 4.38i·10-s + (−0.366 + 0.285i)12-s + (0.526 − 0.725i)13-s + (−4.05 − 1.31i)14-s + (−2.82 + 4.17i)15-s + (3.61 − 2.62i)16-s + (2.11 − 1.53i)17-s + (−4.50 + 0.305i)18-s + ⋯ |
L(s) = 1 | + (0.861 + 0.625i)2-s + (0.341 + 0.940i)3-s + (0.0414 + 0.127i)4-s + (0.764 + 1.05i)5-s + (−0.294 + 1.02i)6-s + (−1.01 + 0.330i)7-s + (0.284 − 0.877i)8-s + (−0.767 + 0.641i)9-s + 1.38i·10-s + (−0.105 + 0.0823i)12-s + (0.146 − 0.201i)13-s + (−1.08 − 0.351i)14-s + (−0.728 + 1.07i)15-s + (0.902 − 0.655i)16-s + (0.511 − 0.371i)17-s + (−1.06 + 0.0719i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29485 + 1.87211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29485 + 1.87211i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.590 - 1.62i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.21 - 0.885i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-1.71 - 2.35i)T + (-1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (2.68 - 0.874i)T + (5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.526 + 0.725i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.11 + 1.53i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.32 - 0.756i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (2.66 + 8.20i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-8.24 - 5.99i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.09 + 3.36i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.39 - 4.29i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 4.24iT - 43T^{2} \) |
| 47 | \( 1 + (-2.02 - 0.658i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.87 + 5.33i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.542 - 0.176i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (5.98 + 8.23i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + (5.58 + 7.69i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.05 + 0.991i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (4.31 - 5.94i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.11 + 3.71i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 12.4iT - 89T^{2} \) |
| 97 | \( 1 + (4.63 + 3.36i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.71935331094482821406325660295, −10.41956950884657729040216066204, −9.952621961719944467110059381398, −9.285706653754103677205165824288, −7.75436755266716850089978300191, −6.48701499184256075000357922960, −5.95309333017009649052780029431, −4.93317103657952528251773041985, −3.58580749273460449560051020835, −2.76666886963905716799163156624,
1.36139087249496347041035081767, 2.73857826280992008119888291090, 3.82236525004929038334152491086, 5.22059019526324723391055235461, 6.09958813415160630262331670093, 7.30141238884934937641330106539, 8.460814327783749308563611848379, 9.220165443274005699295010458438, 10.27904313234624981040991215839, 11.61948460954462553857248026415