| L(s) = 1 | + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.617 − 3.49i)5-s + (−0.244 − 0.205i)7-s + (0.500 − 0.866i)8-s + (−1.77 − 3.07i)10-s + (−0.773 + 4.38i)11-s + (4.39 + 1.60i)13-s + (−0.300 − 0.109i)14-s + (0.173 − 0.984i)16-s + (0.567 + 0.982i)17-s + (−0.928 + 1.60i)19-s + (−2.72 − 2.28i)20-s + (0.773 + 4.38i)22-s + (−0.110 + 0.0926i)23-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.383 − 0.321i)4-s + (−0.275 − 1.56i)5-s + (−0.0925 − 0.0776i)7-s + (0.176 − 0.306i)8-s + (−0.561 − 0.973i)10-s + (−0.233 + 1.32i)11-s + (1.21 + 0.443i)13-s + (−0.0802 − 0.0292i)14-s + (0.0434 − 0.246i)16-s + (0.137 + 0.238i)17-s + (−0.213 + 0.369i)19-s + (−0.608 − 0.510i)20-s + (0.164 + 0.934i)22-s + (−0.0230 + 0.0193i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37040 - 0.730359i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37040 - 0.730359i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.617 + 3.49i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.244 + 0.205i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.773 - 4.38i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.39 - 1.60i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.567 - 0.982i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.928 - 1.60i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.110 - 0.0926i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (4.09 - 1.49i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.514 + 0.431i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.79 + 6.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.04 - 0.744i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.23 - 6.98i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.91 - 6.63i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 0.805T + 53T^{2} \) |
| 59 | \( 1 + (0.517 + 2.93i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.67 + 2.24i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (6.99 + 2.54i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (4.04 + 7.01i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.30 - 12.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.8 + 4.30i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (5.08 - 1.85i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.52 + 4.37i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.24 + 18.3i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.71466121373404654439906401870, −12.06163130308780512316081627601, −10.91306525939837882299533472502, −9.637783624729829551495430835392, −8.679586147252383293036129879272, −7.47814765900931310930926018958, −5.96625878960616884528932471475, −4.78861018298391905678126954744, −3.90995933279111134493845871754, −1.63739275258173341701393103055,
2.88472112685116412134947135781, 3.73620688801197560256892678594, 5.67934625882443467408105351919, 6.49893976630276589747969973201, 7.58430899762155296303026052096, 8.711100055481158769212833980497, 10.49005561460321290454973232752, 11.01568360746577456062406856720, 11.91067949518287788232450594121, 13.42648921850884539623893944856
