| L(s) = 1 | + (−5.21 + 1.89i)5-s + (1.58 + 9.00i)7-s + (67.4 + 24.5i)11-s + (−63.1 − 53.0i)13-s + (−21.0 + 36.4i)17-s + (−22.5 − 39.0i)19-s + (−12.1 + 68.7i)23-s + (−72.1 + 60.5i)25-s + (−142. + 119. i)29-s + (−46.3 + 262. i)31-s + (−25.3 − 43.9i)35-s + (−40.4 + 70.0i)37-s + (216. + 181. i)41-s + (−249. − 90.9i)43-s + (10.5 + 59.8i)47-s + ⋯ |
| L(s) = 1 | + (−0.466 + 0.169i)5-s + (0.0857 + 0.486i)7-s + (1.85 + 0.673i)11-s + (−1.34 − 1.13i)13-s + (−0.299 + 0.519i)17-s + (−0.271 − 0.471i)19-s + (−0.109 + 0.623i)23-s + (−0.577 + 0.484i)25-s + (−0.913 + 0.766i)29-s + (−0.268 + 1.52i)31-s + (−0.122 − 0.212i)35-s + (−0.179 + 0.311i)37-s + (0.825 + 0.692i)41-s + (−0.885 − 0.322i)43-s + (0.0327 + 0.185i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.815i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9910812212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9910812212\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (5.21 - 1.89i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-1.58 - 9.00i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-67.4 - 24.5i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (63.1 + 53.0i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (21.0 - 36.4i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (22.5 + 39.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (12.1 - 68.7i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (142. - 119. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (46.3 - 262. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (40.4 - 70.0i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-216. - 181. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (249. + 90.9i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-10.5 - 59.8i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 206.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (558. - 203. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-124. - 706. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-23.6 - 19.8i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (155. - 269. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (263. + 457. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (500. - 420. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-883. + 741. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (240. + 417. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (858. + 312. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−11.70177506463983697849238348349, −10.59972452874916633125329462819, −9.547898292895879352528202208491, −8.810169475705465622004266123747, −7.56587608490711166118915457247, −6.82855564852983465128155753035, −5.55024611520141620386516430837, −4.38938141575471408328972159988, −3.20103618296361416839375917429, −1.63970486672362177609643827617,
0.35444401047821836483411766448, 2.02537363570648343568326184348, 3.86408917269712300321764374486, 4.45654319065741965665176398433, 6.08608053938887854068726820394, 6.99220734626118980658918396477, 7.942360739515946144019107526666, 9.152227534544237835553765219626, 9.693483798087017667949543065048, 11.12120122117785619748177799471
