| L(s) = 1 | + (0.959 − 0.281i)2-s − 3-s + (0.841 − 0.540i)4-s + (0.564 + 0.651i)5-s + (−0.959 + 0.281i)6-s + (−1.20 − 2.63i)7-s + (0.654 − 0.755i)8-s + 9-s + (0.725 + 0.466i)10-s + (−3.02 − 1.35i)11-s + (−0.841 + 0.540i)12-s + (3.09 − 1.98i)13-s + (−1.89 − 2.18i)14-s + (−0.564 − 0.651i)15-s + (0.415 − 0.909i)16-s + (0.0348 − 0.242i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s − 0.577·3-s + (0.420 − 0.270i)4-s + (0.252 + 0.291i)5-s + (−0.391 + 0.115i)6-s + (−0.454 − 0.994i)7-s + (0.231 − 0.267i)8-s + 0.333·9-s + (0.229 + 0.147i)10-s + (−0.913 − 0.407i)11-s + (−0.242 + 0.156i)12-s + (0.857 − 0.551i)13-s + (−0.506 − 0.584i)14-s + (−0.145 − 0.168i)15-s + (0.103 − 0.227i)16-s + (0.00844 − 0.0587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0222 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0222 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16246 - 1.18859i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16246 - 1.18859i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + (3.02 + 1.35i)T \) |
| good | 5 | \( 1 + (-0.564 - 0.651i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (1.20 + 2.63i)T + (-4.58 + 5.29i)T^{2} \) |
| 13 | \( 1 + (-3.09 + 1.98i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.0348 + 0.242i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.420 + 2.92i)T + (-18.2 + 5.35i)T^{2} \) |
| 23 | \( 1 + (-0.0484 + 0.106i)T + (-15.0 - 17.3i)T^{2} \) |
| 29 | \( 1 + (0.512 + 3.56i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (0.269 + 0.173i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.98 - 2.55i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-4.95 + 1.45i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.00587 - 0.00677i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (6.35 + 1.86i)T + (39.5 + 25.4i)T^{2} \) |
| 53 | \( 1 + (-0.925 - 2.02i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.72 - 1.09i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.304 + 0.0894i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (3.79 - 1.11i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-1.36 - 9.46i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-1.81 + 3.96i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.65 + 4.22i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.356 - 0.781i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (1.35 - 9.44i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (0.861 - 0.994i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.48423688906006393418492367867, −9.721169699054360980695855168986, −8.313136955863325002104586660077, −7.33355862586077634713823662152, −6.43268651696614307156616332090, −5.75482983572590162904799419531, −4.68527488712407640755014634889, −3.68011786673254796205772694522, −2.59430514582215664310705353547, −0.74544902713834515052727180336,
1.81147830118968940972106240790, 3.12372288737944738600623234518, 4.38267395968211372600772320349, 5.42736913349738732541803339181, 5.92114721560616548974029823268, 6.85474876470900873286924083153, 7.924119865108133624428062896686, 8.933474789180443026117045833102, 9.773719318130557735968212657028, 10.82113976344045404338273713606
