| L(s) = 1 | + (−0.939 + 0.342i)3-s + (0.590 − 3.34i)5-s + (1.12 − 0.652i)7-s + (0.766 − 0.642i)9-s + (0.186 + 0.107i)11-s + (0.942 − 2.58i)13-s + (0.590 + 3.34i)15-s + (0.590 + 0.495i)17-s + (4.01 + 1.70i)19-s + (−0.838 + 0.999i)21-s + (−7.01 + 1.23i)23-s + (−6.16 − 2.24i)25-s + (−0.500 + 0.866i)27-s + (−5.04 − 6.00i)29-s + (−2.21 − 3.83i)31-s + ⋯ |
| L(s) = 1 | + (−0.542 + 0.197i)3-s + (0.264 − 1.49i)5-s + (0.427 − 0.246i)7-s + (0.255 − 0.214i)9-s + (0.0562 + 0.0324i)11-s + (0.261 − 0.717i)13-s + (0.152 + 0.864i)15-s + (0.143 + 0.120i)17-s + (0.920 + 0.390i)19-s + (−0.183 + 0.218i)21-s + (−1.46 + 0.257i)23-s + (−1.23 − 0.449i)25-s + (−0.0962 + 0.166i)27-s + (−0.936 − 1.11i)29-s + (−0.397 − 0.688i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.781604 - 0.981264i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.781604 - 0.981264i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (-4.01 - 1.70i)T \) |
| good | 5 | \( 1 + (-0.590 + 3.34i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.12 + 0.652i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.186 - 0.107i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.942 + 2.58i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.590 - 0.495i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (7.01 - 1.23i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.04 + 6.00i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.21 + 3.83i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.08iT - 37T^{2} \) |
| 41 | \( 1 + (-0.000924 - 0.00253i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-10.0 - 1.77i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.0760 - 0.0906i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-2.58 + 0.455i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (4.17 + 3.50i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.475 + 2.69i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.399 + 0.335i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.74 + 15.5i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.35 + 1.94i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (8.45 - 3.07i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (7.27 - 4.20i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.20 + 6.05i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.39 + 1.65i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−9.690571268042495593965835550741, −9.231077429973994578835144648260, −8.037732192194570121328999103162, −7.62476649606713369970011965685, −5.92447415796662072916306857274, −5.60528677324330855782256788786, −4.56618073261648607597288270129, −3.77661433372947138637207224007, −1.86318557854695816093275343817, −0.65646799863114775124619328022,
1.69638606482188493502291031928, 2.85015955559258181676509588816, 3.98267965067002877283420206014, 5.28454593510520413477253552448, 6.09921352279667925045143337913, 6.93396700101914307094410148801, 7.49397923876501352827817829767, 8.662689567845716125950961612926, 9.712357519384677826995458100693, 10.42904286809203380020799934381
