| L(s) = 1 | + (0.938 + 2.84i)3-s + (−4.88 − 1.05i)5-s − 6.81i·7-s + (−7.23 + 5.34i)9-s + 7.52i·11-s + 16.2i·13-s + (−1.58 − 14.9i)15-s + 4.11·17-s − 7.86·19-s + (19.4 − 6.39i)21-s − 19.5·23-s + (22.7 + 10.2i)25-s + (−22.0 − 15.6i)27-s − 55.8i·29-s + 43.4·31-s + ⋯ |
| L(s) = 1 | + (0.312 + 0.949i)3-s + (−0.977 − 0.210i)5-s − 0.973i·7-s + (−0.804 + 0.594i)9-s + 0.684i·11-s + 1.24i·13-s + (−0.105 − 0.994i)15-s + 0.242·17-s − 0.413·19-s + (0.924 − 0.304i)21-s − 0.848·23-s + (0.911 + 0.411i)25-s + (−0.815 − 0.578i)27-s − 1.92i·29-s + 1.40·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6717028776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6717028776\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.938 - 2.84i)T \) |
| 5 | \( 1 + (4.88 + 1.05i)T \) |
| good | 7 | \( 1 + 6.81iT - 49T^{2} \) |
| 11 | \( 1 - 7.52iT - 121T^{2} \) |
| 13 | \( 1 - 16.2iT - 169T^{2} \) |
| 17 | \( 1 - 4.11T + 289T^{2} \) |
| 19 | \( 1 + 7.86T + 361T^{2} \) |
| 23 | \( 1 + 19.5T + 529T^{2} \) |
| 29 | \( 1 + 55.8iT - 841T^{2} \) |
| 31 | \( 1 - 43.4T + 961T^{2} \) |
| 37 | \( 1 + 31.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 61.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 82.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 97.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.13T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 40.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 78.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 51.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 2.72T + 6.88e3T^{2} \) |
| 89 | \( 1 + 70.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 3.44iT - 9.40e3T^{2} \) |
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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
−9.776168959375248994665225805294, −8.813713990447036606809115693980, −8.004060257604525726713709274286, −7.30267323634203612639430830918, −6.24785770123019324675148008123, −4.78682181441687490140279514852, −4.22898658356837337578534992032, −3.63826631767876355339360733961, −2.12657613726709766239178069063, −0.22118345853826438440072569220,
1.20707123119562295836735864718, 2.83782001641511801366556214754, 3.26219709093679832099823368627, 4.83081207115366178491982179256, 5.94365087333689665816217092207, 6.59632170040134875420378763422, 7.77710049740249739172481463759, 8.218320046473231862493677563763, 8.810382318222822264932868912691, 10.02694615360341258417387771170
