| L(s) = 1 | + (3 − 8.48i)3-s + 16.9i·5-s + 26·7-s + (−62.9 − 50.9i)9-s − 118. i·11-s − 50·13-s + (143. + 50.9i)15-s − 203. i·17-s + 358·19-s + (78 − 220. i)21-s − 373. i·23-s + 337.·25-s + (−620. + 381. i)27-s − 1.44e3i·29-s − 742·31-s + ⋯ |
| L(s) = 1 | + (0.333 − 0.942i)3-s + 0.678i·5-s + 0.530·7-s + (−0.777 − 0.628i)9-s − 0.981i·11-s − 0.295·13-s + (0.639 + 0.226i)15-s − 0.704i·17-s + 0.991·19-s + (0.176 − 0.500i)21-s − 0.705i·23-s + 0.539·25-s + (−0.851 + 0.523i)27-s − 1.71i·29-s − 0.772·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.818353534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.818353534\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3 + 8.48i)T \) |
| good | 5 | \( 1 - 16.9iT - 625T^{2} \) |
| 7 | \( 1 - 26T + 2.40e3T^{2} \) |
| 11 | \( 1 + 118. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 50T + 2.85e4T^{2} \) |
| 17 | \( 1 + 203. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 358T + 1.30e5T^{2} \) |
| 23 | \( 1 + 373. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.44e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 742T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.87e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.40e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 262T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.69e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 458. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.81e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.48e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.48e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.56e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 290T + 2.83e7T^{2} \) |
| 79 | \( 1 - 9.81e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.11e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.84e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 478T + 8.85e7T^{2} \) |
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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
−11.59214694489718503583264401374, −10.82882629240916553376013116433, −9.426504158342018505122057424575, −8.340427143598179235707095022318, −7.45126370185651206957102741019, −6.51861008995150278593856158806, −5.31960352023633933881650303822, −3.42911193892990786501886081777, −2.29090268279638721015791439585, −0.63398851782687898838675883218,
1.66263590166931587735789856654, 3.40291170043334504149268685203, 4.72280031133293900194793702409, 5.35627037593822518407689628404, 7.20920406904615737950802354198, 8.339522437574098527424032689293, 9.209624262902816517874453476204, 10.07486325947133788879535089804, 11.06977077591468786882603903159, 12.13572084323587795842806660455
