| L(s) = 1 | + (−0.186 + 0.982i)2-s + (2.60 + 0.0610i)3-s + (−0.930 − 0.366i)4-s + (0.0581 + 0.220i)5-s + (−0.545 + 2.54i)6-s + (2.03 − 4.54i)7-s + (0.533 − 0.845i)8-s + (3.77 + 0.177i)9-s + (−0.227 + 0.0160i)10-s + (−3.64 + 4.09i)11-s + (−2.40 − 1.01i)12-s + (0.357 + 5.07i)13-s + (4.08 + 2.85i)14-s + (0.137 + 0.577i)15-s + (0.731 + 0.681i)16-s + (5.65 − 5.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.131 + 0.694i)2-s + (1.50 + 0.0352i)3-s + (−0.465 − 0.183i)4-s + (0.0259 + 0.0985i)5-s + (−0.222 + 1.03i)6-s + (0.770 − 1.71i)7-s + (0.188 − 0.299i)8-s + (1.25 + 0.0590i)9-s + (−0.0719 + 0.00506i)10-s + (−1.09 + 1.23i)11-s + (−0.692 − 0.291i)12-s + (0.0990 + 1.40i)13-s + (1.09 + 0.761i)14-s + (0.0356 + 0.149i)15-s + (0.182 + 0.170i)16-s + (1.37 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.12664 + 0.632675i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12664 + 0.632675i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.186 - 0.982i)T \) |
| 269 | \( 1 + (-13.5 + 9.17i)T \) |
| good | 3 | \( 1 + (-2.60 - 0.0610i)T + (2.99 + 0.140i)T^{2} \) |
| 5 | \( 1 + (-0.0581 - 0.220i)T + (-4.34 + 2.46i)T^{2} \) |
| 7 | \( 1 + (-2.03 + 4.54i)T + (-4.65 - 5.23i)T^{2} \) |
| 11 | \( 1 + (3.64 - 4.09i)T + (-1.28 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.357 - 5.07i)T + (-12.8 + 1.82i)T^{2} \) |
| 17 | \( 1 + (-5.65 + 5.02i)T + (1.98 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.17 + 1.44i)T + (7.37 - 17.5i)T^{2} \) |
| 23 | \( 1 + (0.0595 - 2.54i)T + (-22.9 - 1.07i)T^{2} \) |
| 29 | \( 1 + (1.19 - 2.11i)T + (-14.8 - 24.8i)T^{2} \) |
| 31 | \( 1 + (-3.77 + 0.445i)T + (30.1 - 7.20i)T^{2} \) |
| 37 | \( 1 + (3.66 + 6.83i)T + (-20.4 + 30.8i)T^{2} \) |
| 41 | \( 1 + (6.87 - 1.30i)T + (38.1 - 15.0i)T^{2} \) |
| 43 | \( 1 + (-2.98 - 1.69i)T + (22.0 + 36.8i)T^{2} \) |
| 47 | \( 1 + (9.52 - 2.75i)T + (39.7 - 25.0i)T^{2} \) |
| 53 | \( 1 + (5.16 - 7.04i)T + (-15.9 - 50.5i)T^{2} \) |
| 59 | \( 1 + (0.0826 - 0.209i)T + (-43.1 - 40.2i)T^{2} \) |
| 61 | \( 1 + (3.69 - 4.56i)T + (-12.7 - 59.6i)T^{2} \) |
| 67 | \( 1 + (4.98 - 1.96i)T + (49.0 - 45.6i)T^{2} \) |
| 71 | \( 1 + (2.74 + 3.93i)T + (-24.4 + 66.6i)T^{2} \) |
| 73 | \( 1 + (2.57 + 7.03i)T + (-55.6 + 47.2i)T^{2} \) |
| 79 | \( 1 + (12.8 - 9.86i)T + (20.1 - 76.3i)T^{2} \) |
| 83 | \( 1 + (-0.0249 + 0.265i)T + (-81.5 - 15.4i)T^{2} \) |
| 89 | \( 1 + (-2.54 + 15.3i)T + (-84.2 - 28.6i)T^{2} \) |
| 97 | \( 1 + (-4.81 - 5.94i)T + (-20.3 + 94.8i)T^{2} \) |
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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
−10.48934499786636120060919263854, −9.824111993791754033916288459455, −9.092050226004085855891539991975, −7.917035725661495681054335807769, −7.45666886987680683063095131197, −6.97906597775417665858623728866, −4.97473468462238770467261441417, −4.32914289266238749647128909573, −3.09767211729463070235768953733, −1.55697267392664324392981868514,
1.65664510616724219013248683941, 3.00263538881983016583255022324, 3.18903249976758965491158255067, 5.13465098095242025706639940698, 5.79252218678074681499777909690, 7.985606277205071590804415695245, 8.234508053813951231870698265782, 8.670428294754028116536418642694, 9.844178075047552226089086737445, 10.57430278102301203550662515892
