| L(s) = 1 | + (−1.93 − 2.84i)2-s + (−0.162 + 2.99i)3-s + (−1.42 + 3.56i)4-s + (−12.2 + 5.65i)5-s + (8.84 − 5.32i)6-s + (8.68 − 31.2i)7-s + (−13.9 + 3.07i)8-s + (−8.94 − 0.973i)9-s + (39.6 + 23.8i)10-s + (25.9 − 24.5i)11-s + (−10.4 − 4.83i)12-s + (−8.20 + 0.892i)13-s + (−105. + 35.6i)14-s + (−14.9 − 37.5i)15-s + (58.0 + 54.9i)16-s + (29.2 + 105. i)17-s + ⋯ |
| L(s) = 1 | + (−0.682 − 1.00i)2-s + (−0.0312 + 0.576i)3-s + (−0.177 + 0.445i)4-s + (−1.09 + 0.505i)5-s + (0.601 − 0.362i)6-s + (0.469 − 1.68i)7-s + (−0.617 + 0.135i)8-s + (−0.331 − 0.0360i)9-s + (1.25 + 0.755i)10-s + (0.710 − 0.673i)11-s + (−0.251 − 0.116i)12-s + (−0.175 + 0.0190i)13-s + (−2.02 + 0.681i)14-s + (−0.257 − 0.645i)15-s + (0.907 + 0.859i)16-s + (0.417 + 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.300021 + 0.203607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.300021 + 0.203607i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.162 - 2.99i)T \) |
| 59 | \( 1 + (452. - 23.8i)T \) |
| good | 2 | \( 1 + (1.93 + 2.84i)T + (-2.96 + 7.43i)T^{2} \) |
| 5 | \( 1 + (12.2 - 5.65i)T + (80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (-8.68 + 31.2i)T + (-293. - 176. i)T^{2} \) |
| 11 | \( 1 + (-25.9 + 24.5i)T + (72.0 - 1.32e3i)T^{2} \) |
| 13 | \( 1 + (8.20 - 0.892i)T + (2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (-29.2 - 105. i)T + (-4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (86.3 - 65.6i)T + (1.83e3 - 6.60e3i)T^{2} \) |
| 23 | \( 1 + (81.9 - 154. i)T + (-6.82e3 - 1.00e4i)T^{2} \) |
| 29 | \( 1 + (45.9 - 67.7i)T + (-9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (80.9 + 61.5i)T + (7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (-257. - 56.6i)T + (4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (-154. - 290. i)T + (-3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (23.7 + 22.4i)T + (4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (362. + 167. i)T + (6.72e4 + 7.91e4i)T^{2} \) |
| 53 | \( 1 + (-517. + 311. i)T + (6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (-416. - 613. i)T + (-8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (400. - 88.0i)T + (2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (691. + 320. i)T + (2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (-146. + 49.2i)T + (3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (-37.0 - 683. i)T + (-4.90e5 + 5.33e4i)T^{2} \) |
| 83 | \( 1 + (108. + 663. i)T + (-5.41e5 + 1.82e5i)T^{2} \) |
| 89 | \( 1 + (244. - 359. i)T + (-2.60e5 - 6.54e5i)T^{2} \) |
| 97 | \( 1 + (-109. - 36.9i)T + (7.26e5 + 5.52e5i)T^{2} \) |
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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
−11.81846784230212326594344297260, −11.20138232732703996535343709173, −10.55796257283693048658098034819, −9.842864009479517895054018698457, −8.414726738860720109964359604753, −7.63931450823580834506590079316, −6.12266767847120168415234285982, −3.97184911006689509444439506865, −3.62896338942032598056357057857, −1.37045720819124059211961087476,
0.22016455432138487474514823726, 2.50483866668432243791313576719, 4.61634134023297884938687844460, 5.93227011127074664605733825431, 7.04278196844110028745193062761, 7.973181428956717331987922022423, 8.716019307235728890283118826727, 9.388700238152457909094595745117, 11.44351626086276933469746879625, 12.14250408767088997699354207969
