| L(s) = 1 | + (3.23 − 2.35i)2-s + (4.94 − 15.2i)4-s + (42.4 − 58.4i)5-s + (−51.0 − 16.5i)7-s + (−19.7 − 60.8i)8-s − 288. i·10-s + (342. − 209. i)11-s + (−203. − 279. i)13-s + (−204. + 66.3i)14-s + (−207. − 150. i)16-s + (29.4 + 21.4i)17-s + (−753. + 244. i)19-s + (−679. − 935. i)20-s + (614. − 1.48e3i)22-s − 2.10e3i·23-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (0.759 − 1.04i)5-s + (−0.393 − 0.127i)7-s + (−0.109 − 0.336i)8-s − 0.913i·10-s + (0.852 − 0.522i)11-s + (−0.333 − 0.458i)13-s + (−0.278 + 0.0904i)14-s + (−0.202 − 0.146i)16-s + (0.0247 + 0.0179i)17-s + (−0.478 + 0.155i)19-s + (−0.379 − 0.522i)20-s + (0.270 − 0.653i)22-s − 0.828i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.616510322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.616510322\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-342. + 209. i)T \) |
| good | 5 | \( 1 + (-42.4 + 58.4i)T + (-965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (51.0 + 16.5i)T + (1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (203. + 279. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-29.4 - 21.4i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (753. - 244. i)T + (2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + 2.10e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (929. - 2.86e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (886. - 644. i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-368. + 1.13e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (185. + 570. i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.26e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (5.05e3 - 1.64e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (6.82e3 + 9.39e3i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (3.91e4 + 1.27e4i)T + (5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-2.55e4 + 3.51e4i)T + (-2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 - 2.74e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (4.87e4 - 6.71e4i)T + (-5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-4.79e4 - 1.55e4i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.27e3 - 3.13e3i)T + (-9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-7.21e3 - 5.24e3i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 - 1.86e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.35e4 + 2.44e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−11.31351750934547972715993880428, −10.21116509477670191431142980258, −9.325662792861673331085405522873, −8.435496306317815857280922039283, −6.73926669564378690118875928945, −5.73576127220949547666731856124, −4.74720628830294188865267826639, −3.45970913002522928748867161737, −1.89705967288222926003193941532, −0.63316242021303209723617368445,
1.95488867049798629122983377131, 3.17903656232004000019215433335, 4.50352537995823100051686861311, 5.99696886874618352086340385795, 6.62155231704063264377114816900, 7.57997739017632833894470446405, 9.169955874800703203371598768068, 9.945708834178152756013560930220, 11.12701912591694310320091628520, 12.06337502613972691912747634266
