| L(s) = 1 | + (3.23 + 2.35i)2-s + (4.94 + 15.2i)4-s + (−0.439 − 0.604i)5-s + (195. − 63.4i)7-s + (−19.7 + 60.8i)8-s − 2.98i·10-s + (−226. + 331. i)11-s + (−80.6 + 110. i)13-s + (781. + 253. i)14-s + (−207. + 150. i)16-s + (966. − 702. i)17-s + (1.44e3 + 468. i)19-s + (7.02 − 9.67i)20-s + (−1.51e3 + 538. i)22-s − 3.05e3i·23-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.00785 − 0.0108i)5-s + (1.50 − 0.489i)7-s + (−0.109 + 0.336i)8-s − 0.00945i·10-s + (−0.565 + 0.825i)11-s + (−0.132 + 0.182i)13-s + (1.06 + 0.346i)14-s + (−0.202 + 0.146i)16-s + (0.811 − 0.589i)17-s + (0.916 + 0.297i)19-s + (0.00392 − 0.00540i)20-s + (−0.666 + 0.237i)22-s − 1.20i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.323388539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.323388539\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (226. - 331. i)T \) |
| good | 5 | \( 1 + (0.439 + 0.604i)T + (-965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-195. + 63.4i)T + (1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (80.6 - 110. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-966. + 702. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.44e3 - 468. i)T + (2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + 3.05e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-2.05e3 - 6.33e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-4.24e3 - 3.08e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.15e3 - 9.72e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (4.56e3 - 1.40e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 7.97e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-8.14e3 - 2.64e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.24e4 - 1.70e4i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.37e3 + 446. i)T + (5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (2.93e4 + 4.03e4i)T + (-2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 - 5.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-1.24e4 - 1.71e4i)T + (-5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-2.93e4 + 9.52e3i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-6.34e4 + 8.72e4i)T + (-9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (7.59e4 - 5.51e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + 6.42e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (6.16e4 + 4.48e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−11.92213745770239903259033995628, −10.87440008528940298564788080148, −9.890140714107646983122324235038, −8.316813611053078527468718386883, −7.69021019201855260825837283176, −6.62611114356013897027661037159, −5.00012497187075805973916703042, −4.63113356958231030332018329315, −2.87061897713230985545599375657, −1.28081774728355370295551481129,
1.02023139022920173480202957229, 2.35628660177055031709658263303, 3.73042600419895458375177953796, 5.19185900390053609857110975796, 5.70459195574155303240606081852, 7.51615013582824593916596721189, 8.331537331151469951880835994818, 9.607959920175861056276220654795, 10.81916642088879415746754798732, 11.48675060040116004551934689596
