| 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
−10.57430278102301203550662515892, −9.844178075047552226089086737445, −8.670428294754028116536418642694, −8.234508053813951231870698265782, −7.985606277205071590804415695245, −5.79252218678074681499777909690, −5.13465098095242025706639940698, −3.18903249976758965491158255067, −3.00263538881983016583255022324, −1.65664510616724219013248683941,
1.55697267392664324392981868514, 3.09767211729463070235768953733, 4.32914289266238749647128909573, 4.97473468462238770467261441417, 6.97906597775417665858623728866, 7.45666886987680683063095131197, 7.917035725661495681054335807769, 9.092050226004085855891539991975, 9.824111993791754033916288459455, 10.48934499786636120060919263854
