| L(s) = 1 | + (−1.5 + 0.866i)3-s + (1.77 + 1.02i)5-s + (6.69 + 2.05i)7-s + (1.5 − 2.59i)9-s + (−6.13 − 10.6i)11-s − 16.5i·13-s − 3.54·15-s + (5.61 − 3.24i)17-s + (−11.2 − 6.46i)19-s + (−11.8 + 2.71i)21-s + (−13.4 + 23.2i)23-s + (−10.4 − 18.0i)25-s + 5.19i·27-s + 38.4·29-s + (−6.60 + 3.81i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 + 0.288i)3-s + (0.354 + 0.204i)5-s + (0.956 + 0.293i)7-s + (0.166 − 0.288i)9-s + (−0.557 − 0.966i)11-s − 1.27i·13-s − 0.236·15-s + (0.330 − 0.190i)17-s + (−0.589 − 0.340i)19-s + (−0.562 + 0.129i)21-s + (−0.583 + 1.01i)23-s + (−0.416 − 0.721i)25-s + 0.192i·27-s + 1.32·29-s + (−0.213 + 0.123i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.515563797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.515563797\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 7 | \( 1 + (-6.69 - 2.05i)T \) |
| good | 5 | \( 1 + (-1.77 - 1.02i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (6.13 + 10.6i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 16.5iT - 169T^{2} \) |
| 17 | \( 1 + (-5.61 + 3.24i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (11.2 + 6.46i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (13.4 - 23.2i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 38.4T + 841T^{2} \) |
| 31 | \( 1 + (6.60 - 3.81i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-26.9 + 46.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 6.44iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 13.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + (24.4 + 14.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (34.4 + 59.6i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-62.8 + 36.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (21.4 + 12.4i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (35.3 + 61.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 12.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-93.9 + 54.2i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-48.8 + 84.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 83.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-51.6 - 29.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 147. iT - 9.40e3T^{2} \) |
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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
−10.41423777151292697290879197252, −9.415531775430922628510399512895, −8.266403057084979527839949434958, −7.78863540246330246899306450627, −6.34515953540988975579728827798, −5.58178064163041949158427834033, −4.90032280952655646923419837733, −3.52440105262091064511316738397, −2.28818401272860710732999063025, −0.61800376211319404647286818336,
1.35888604270088618808324550186, 2.30547707554119720442351375944, 4.28444953859815574719825604558, 4.81804170060541373958980186270, 5.98450723810829380565615921362, 6.87714838698119734097720573006, 7.78853180277460425041352345871, 8.581583737375498751148315608145, 9.756434083600523865418266372672, 10.41778523120359726759886202762
