| L(s) = 1 | + (−5.17 − 1.68i)2-s + (−12.6 + 9.16i)3-s + (−27.8 − 20.2i)4-s + (−25.6 − 78.9i)5-s + (80.6 − 26.1i)6-s + (−29.3 + 40.4i)7-s + (314. + 432. i)8-s + (75.0 − 231. i)9-s + 451. i·10-s + (−670. + 1.14e3i)11-s + 536.·12-s + (3.71e3 + 1.20e3i)13-s + (219. − 159. i)14-s + (1.04e3 + 760. i)15-s + (−218. − 671. i)16-s + (3.27e3 − 1.06e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.646 − 0.210i)2-s + (−0.467 + 0.339i)3-s + (−0.435 − 0.316i)4-s + (−0.205 − 0.631i)5-s + (0.373 − 0.121i)6-s + (−0.0856 + 0.117i)7-s + (0.614 + 0.845i)8-s + (0.103 − 0.317i)9-s + 0.451i·10-s + (−0.504 + 0.863i)11-s + 0.310·12-s + (1.69 + 0.549i)13-s + (0.0801 − 0.0582i)14-s + (0.310 + 0.225i)15-s + (−0.0532 − 0.163i)16-s + (0.665 − 0.216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.685689 + 0.252765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.685689 + 0.252765i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (12.6 - 9.16i)T \) |
| 11 | \( 1 + (670. - 1.14e3i)T \) |
| good | 2 | \( 1 + (5.17 + 1.68i)T + (51.7 + 37.6i)T^{2} \) |
| 5 | \( 1 + (25.6 + 78.9i)T + (-1.26e4 + 9.18e3i)T^{2} \) |
| 7 | \( 1 + (29.3 - 40.4i)T + (-3.63e4 - 1.11e5i)T^{2} \) |
| 13 | \( 1 + (-3.71e3 - 1.20e3i)T + (3.90e6 + 2.83e6i)T^{2} \) |
| 17 | \( 1 + (-3.27e3 + 1.06e3i)T + (1.95e7 - 1.41e7i)T^{2} \) |
| 19 | \( 1 + (-3.32e3 - 4.57e3i)T + (-1.45e7 + 4.47e7i)T^{2} \) |
| 23 | \( 1 + 1.54e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.95e4 - 2.68e4i)T + (-1.83e8 - 5.65e8i)T^{2} \) |
| 31 | \( 1 + (-7.25e3 + 2.23e4i)T + (-7.18e8 - 5.21e8i)T^{2} \) |
| 37 | \( 1 + (-5.27e4 - 3.83e4i)T + (7.92e8 + 2.44e9i)T^{2} \) |
| 41 | \( 1 + (-2.27e4 - 3.12e4i)T + (-1.46e9 + 4.51e9i)T^{2} \) |
| 43 | \( 1 - 5.68e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-3.10e4 + 2.25e4i)T + (3.33e9 - 1.02e10i)T^{2} \) |
| 53 | \( 1 + (2.84e4 - 8.75e4i)T + (-1.79e10 - 1.30e10i)T^{2} \) |
| 59 | \( 1 + (-1.06e5 - 7.71e4i)T + (1.30e10 + 4.01e10i)T^{2} \) |
| 61 | \( 1 + (-2.49e5 + 8.12e4i)T + (4.16e10 - 3.02e10i)T^{2} \) |
| 67 | \( 1 + 4.94e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-1.07e5 - 3.31e5i)T + (-1.03e11 + 7.52e10i)T^{2} \) |
| 73 | \( 1 + (2.79e4 - 3.85e4i)T + (-4.67e10 - 1.43e11i)T^{2} \) |
| 79 | \( 1 + (3.41e5 + 1.10e5i)T + (1.96e11 + 1.42e11i)T^{2} \) |
| 83 | \( 1 + (2.08e5 - 6.78e4i)T + (2.64e11 - 1.92e11i)T^{2} \) |
| 89 | \( 1 + 1.98e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (1.94e5 - 5.97e5i)T + (-6.73e11 - 4.89e11i)T^{2} \) |
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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
−15.92534413459972992641931451995, −14.39297047805814874845186093551, −13.02028194275645311390932237337, −11.62582939052584151777746445689, −10.33508296253618727363562257477, −9.284372598977341333531008012969, −8.027236144073472710873412083221, −5.77644537509614884081081295443, −4.32109129557625160241886391005, −1.22452632533298438277085213343,
0.63053833828956303684517764282, 3.58560424864041582389397539908, 5.90065679566135800413858400159, 7.48788920683700467524661437713, 8.564942046352755249186150042107, 10.26582478471064750597805896652, 11.33532710637010807809957707507, 12.98304227465983482804703038716, 13.85630766369589415540145056967, 15.71490730547139070001221018769
