| L(s) = 1 | + (−2.98 + 0.291i)3-s − 4.46i·7-s + (8.82 − 1.74i)9-s − 17.8i·11-s + 11.0i·13-s + 0.794·17-s + 26.5·19-s + (1.30 + 13.3i)21-s + 14.9·23-s + (−25.8 + 7.77i)27-s + 5.58i·29-s − 53.1·31-s + (5.21 + 53.3i)33-s + 51.7i·37-s + (−3.21 − 32.8i)39-s + ⋯ |
| L(s) = 1 | + (−0.995 + 0.0972i)3-s − 0.637i·7-s + (0.981 − 0.193i)9-s − 1.62i·11-s + 0.846i·13-s + 0.0467·17-s + 1.39·19-s + (0.0619 + 0.634i)21-s + 0.649·23-s + (−0.957 + 0.287i)27-s + 0.192i·29-s − 1.71·31-s + (0.157 + 1.61i)33-s + 1.39i·37-s + (−0.0823 − 0.842i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9944641123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9944641123\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.98 - 0.291i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.46iT - 49T^{2} \) |
| 11 | \( 1 + 17.8iT - 121T^{2} \) |
| 13 | \( 1 - 11.0iT - 169T^{2} \) |
| 17 | \( 1 - 0.794T + 289T^{2} \) |
| 19 | \( 1 - 26.5T + 361T^{2} \) |
| 23 | \( 1 - 14.9T + 529T^{2} \) |
| 29 | \( 1 - 5.58iT - 841T^{2} \) |
| 31 | \( 1 + 53.1T + 961T^{2} \) |
| 37 | \( 1 - 51.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 67.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 12.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 37.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 61.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 97.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 3.02iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 57.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 31.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 2.16T + 6.24e3T^{2} \) |
| 83 | \( 1 + 13.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 173. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 91.6iT - 9.40e3T^{2} \) |
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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
−9.364190400793446979268814357309, −8.597015231052243344986779144021, −7.36104613087824953418440947958, −6.87545318531925687404487913066, −5.78611219300548726954803208541, −5.24247445100511532191447605006, −4.06822555378816981248469304217, −3.26300769675066752852680766222, −1.44360508396932744329782913348, −0.39152991276424670778920824537,
1.19984954689582636090295158294, 2.41398814033104868826056082608, 3.80295060961087714038773455540, 5.05123039799856775872547746339, 5.36531701220269703361230249542, 6.46574914251270851406545536910, 7.32587142540753422056437904704, 7.889363930356477702694241688103, 9.365351296069750272845095617322, 9.701894619313360997910499109572
