| L(s) = 1 | + (−1.28 + 2.71i)3-s + (0.454 − 0.787i)5-s + (6.10 − 3.52i)7-s + (−5.68 − 6.97i)9-s + (−6.96 + 4.02i)11-s + (−3.35 + 5.81i)13-s + (1.54 + 2.24i)15-s − 26.3·17-s + 20.5i·19-s + (1.69 + 21.0i)21-s + (21.8 + 12.6i)23-s + (12.0 + 20.9i)25-s + (26.2 − 6.44i)27-s + (−15.1 − 26.2i)29-s + (0.120 + 0.0693i)31-s + ⋯ |
| L(s) = 1 | + (−0.428 + 0.903i)3-s + (0.0909 − 0.157i)5-s + (0.872 − 0.503i)7-s + (−0.632 − 0.774i)9-s + (−0.633 + 0.365i)11-s + (−0.258 + 0.447i)13-s + (0.103 + 0.149i)15-s − 1.54·17-s + 1.08i·19-s + (0.0809 + 1.00i)21-s + (0.949 + 0.547i)23-s + (0.483 + 0.837i)25-s + (0.971 − 0.238i)27-s + (−0.523 − 0.906i)29-s + (0.00387 + 0.00223i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6296223784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6296223784\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.28 - 2.71i)T \) |
| good | 5 | \( 1 + (-0.454 + 0.787i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-6.10 + 3.52i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (6.96 - 4.02i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (3.35 - 5.81i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 26.3T + 289T^{2} \) |
| 19 | \( 1 - 20.5iT - 361T^{2} \) |
| 23 | \( 1 + (-21.8 - 12.6i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (15.1 + 26.2i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-0.120 - 0.0693i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 69.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (29.3 - 50.8i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-2.45 + 1.41i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (70.7 - 40.8i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 30.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (77.1 + 44.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (24.0 + 41.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-44.0 - 25.4i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 68.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 22.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-34.4 + 19.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (23.0 - 13.3i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 25.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + (52.3 + 90.7i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.03669412808310846130673106115, −10.10712599395262253677538882106, −9.306871825374588216591752422570, −8.408347856314429436243366637819, −7.37208640445804666226503825208, −6.30550429101066226749053043341, −5.03944372812790581452618465518, −4.62589181529727155369203062900, −3.40657423486086845738397001699, −1.73070795743777027012719680342,
0.23964662389476282460368279841, 1.91252698584032648909727945977, 2.85968794258280207756524747140, 4.84010121942245275808515943468, 5.36320527119498812886871017109, 6.65178773812383757984839624140, 7.22489692213205695488742591846, 8.510293297006095539604546921445, 8.764437277650002315797899619533, 10.52699988027553159184747318471
