| L(s) = 1 | + (−1.60 − 2.53i)3-s + (8.13 + 4.69i)7-s + (−3.82 + 8.14i)9-s + (15.4 + 8.90i)11-s + (−11.9 + 6.92i)13-s − 30.7·17-s − 28.5·19-s + (−1.19 − 28.1i)21-s + (−6.16 − 10.6i)23-s + (26.7 − 3.42i)27-s + (5.02 + 2.90i)29-s + (−3.25 − 5.63i)31-s + (−2.26 − 53.3i)33-s − 66.5i·37-s + (36.8 + 19.2i)39-s + ⋯ |
| L(s) = 1 | + (−0.536 − 0.844i)3-s + (1.16 + 0.670i)7-s + (−0.424 + 0.905i)9-s + (1.40 + 0.809i)11-s + (−0.922 + 0.532i)13-s − 1.80·17-s − 1.50·19-s + (−0.0568 − 1.34i)21-s + (−0.267 − 0.464i)23-s + (0.991 − 0.126i)27-s + (0.173 + 0.100i)29-s + (−0.104 − 0.181i)31-s + (−0.0686 − 1.61i)33-s − 1.79i·37-s + (0.943 + 0.492i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3441541725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3441541725\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.60 + 2.53i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-8.13 - 4.69i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-15.4 - 8.90i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (11.9 - 6.92i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 30.7T + 289T^{2} \) |
| 19 | \( 1 + 28.5T + 361T^{2} \) |
| 23 | \( 1 + (6.16 + 10.6i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-5.02 - 2.90i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (3.25 + 5.63i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 66.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (33.0 - 19.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (47.7 + 27.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-8.23 + 14.2i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 69.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (91.2 - 52.6i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (33.5 - 58.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (39.8 - 22.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 31.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 73.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-47.3 + 81.9i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (7.73 - 13.4i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 52.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-66.0 - 38.1i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.48095978752861913727872046499, −9.081681241368677005140755235934, −8.675298637687528875036128964963, −7.56778556739491265182012367784, −6.75185037849118050972146067013, −6.15810684903277229199441381550, −4.80574197415294827655469575945, −4.37602759448980235358470255706, −2.15164323492597164877010912197, −1.84140268751423261615535287466,
0.11272096549677222673588964505, 1.66023382756995377644184311172, 3.34512932336679575125618059124, 4.50062658022668170086806839358, 4.75694385298596299377795694719, 6.20092961505214017178596886766, 6.75188284066880107805949751822, 8.117473748324239385780224604814, 8.764978665708201474574317808145, 9.654572465086463966756576622871
