| L(s) = 1 | + (1.5 + 2.59i)3-s + (1.87 + 3.24i)7-s + (−4.5 + 7.79i)9-s + (−10.1 + 5.84i)11-s + (−1.12 + 1.95i)13-s − 11.6i·17-s − 26.7·19-s + (−5.61 + 9.73i)21-s + (17.2 + 9.95i)23-s − 27·27-s + (−38.2 + 22.0i)29-s + (26.1 − 45.2i)31-s + (−30.3 − 17.5i)33-s − 14·37-s − 6.76·39-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.866i)3-s + (0.267 + 0.463i)7-s + (−0.5 + 0.866i)9-s + (−0.919 + 0.531i)11-s + (−0.0866 + 0.150i)13-s − 0.687i·17-s − 1.40·19-s + (−0.267 + 0.463i)21-s + (0.749 + 0.432i)23-s − 27-s + (−1.31 + 0.761i)29-s + (0.842 − 1.45i)31-s + (−0.919 − 0.531i)33-s − 0.378·37-s − 0.173·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9198088385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9198088385\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-1.87 - 3.24i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (10.1 - 5.84i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (1.12 - 1.95i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 11.6iT - 289T^{2} \) |
| 19 | \( 1 + 26.7T + 361T^{2} \) |
| 23 | \( 1 + (-17.2 - 9.95i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (38.2 - 22.0i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-26.1 + 45.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 14T + 1.36e3T^{2} \) |
| 41 | \( 1 + (22.5 + 12.9i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-20.9 - 36.3i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (39.3 - 22.7i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 10.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (31.8 + 18.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-22.6 - 39.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (49.9 - 86.5i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 13.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (28.3 + 49.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-78 + 45.0i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 95.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (50.8 + 88.0i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.26917183841799730159409887117, −9.498083719626486615644208983494, −8.781590358314302379338887376645, −7.979273446273132529640243656525, −7.13285346863000147123798736211, −5.77625336985699167514072422030, −4.97056562165564338275791748654, −4.18230861928418070100160937410, −2.91963804336873860823203201221, −2.06111382465376767941664283652,
0.25716528976100095628024961561, 1.68694719883067815938096608480, 2.78543258921820133796550094277, 3.85096056184315799706683157028, 5.09921455488945899982913128115, 6.19813867227101876376535716223, 6.93625202827040468967225724699, 7.955696522884177323439065472997, 8.368203079267053581324175164008, 9.279023521788743509590088240182
