| L(s) = 1 | + (1.22 − 0.707i)2-s + (2.94 + 0.548i)3-s + (0.999 − 1.73i)4-s + (−7.34 + 4.24i)5-s + (3.99 − 1.41i)6-s + (−3.5 − 6.06i)7-s − 2.82i·8-s + (8.39 + 3.23i)9-s + (−6 + 10.3i)10-s + (3.89 − 4.56i)12-s − 13-s + (−8.57 − 4.94i)14-s + (−23.9 + 8.48i)15-s + (−2.00 − 3.46i)16-s + (−7.34 − 4.24i)17-s + (12.5 − 1.97i)18-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.983 + 0.182i)3-s + (0.249 − 0.433i)4-s + (−1.46 + 0.848i)5-s + (0.666 − 0.235i)6-s + (−0.5 − 0.866i)7-s − 0.353i·8-s + (0.933 + 0.359i)9-s + (−0.600 + 1.03i)10-s + (0.324 − 0.380i)12-s − 0.0769·13-s + (−0.612 − 0.353i)14-s + (−1.59 + 0.565i)15-s + (−0.125 − 0.216i)16-s + (−0.432 − 0.249i)17-s + (0.698 − 0.109i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.52571 - 0.189495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52571 - 0.189495i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (-2.94 - 0.548i)T \) |
| 7 | \( 1 + (3.5 + 6.06i)T \) |
| good | 5 | \( 1 + (7.34 - 4.24i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + T + 169T^{2} \) |
| 17 | \( 1 + (7.34 + 4.24i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-15.5 - 26.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-7.34 + 4.24i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 16.9iT - 841T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 31T + 1.84e3T^{2} \) |
| 47 | \( 1 + (36.7 - 21.2i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-22.0 - 12.7i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (7.34 + 4.24i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (25 + 43.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (32.5 - 56.2i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 59.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-48.5 + 84.0i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-51.5 - 89.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 42.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (102. - 59.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 166T + 9.40e3T^{2} \) |
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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
−15.47997062696095382674524817076, −14.54902290546885444629092403510, −13.63197705392624282989371629591, −12.27339210981915117284860218310, −10.98021897421775780286997182345, −9.890114605938989129512380723679, −7.957081226234621029761689100565, −6.95392118258012913460775551043, −4.15855011693847804545356155655, −3.21653654134617585231728888020,
3.24154603880256607681780776282, 4.78094676347487696143398577530, 7.01940340624589896308250688239, 8.273093790620669916874206435250, 9.174208608448382497441165164075, 11.57349151106931504158345369019, 12.55342379585005309656141366091, 13.39859748304795897179131686081, 15.01968345690403589806622748369, 15.55364210706679785658933657500
