L(s) = 1 | + (−1.96 + 0.362i)2-s + (3.73 − 1.42i)4-s + (−1.69 − 1.69i)5-s − 5.74·7-s + (−6.83 + 4.16i)8-s + (3.95 + 2.72i)10-s + (5.59 − 5.59i)11-s + (−13.5 + 13.5i)13-s + (11.2 − 2.08i)14-s + (11.9 − 10.6i)16-s − 19.7·17-s + (−21.6 − 21.6i)19-s + (−8.77 − 3.92i)20-s + (−8.97 + 13.0i)22-s − 24.9·23-s + ⋯ |
L(s) = 1 | + (−0.983 + 0.181i)2-s + (0.934 − 0.356i)4-s + (−0.339 − 0.339i)5-s − 0.820·7-s + (−0.853 + 0.520i)8-s + (0.395 + 0.272i)10-s + (0.508 − 0.508i)11-s + (−1.04 + 1.04i)13-s + (0.806 − 0.148i)14-s + (0.745 − 0.666i)16-s − 1.15·17-s + (−1.14 − 1.14i)19-s + (−0.438 − 0.196i)20-s + (−0.407 + 0.592i)22-s − 1.08·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0223078 - 0.131029i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0223078 - 0.131029i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.96 - 0.362i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.69 + 1.69i)T + 25iT^{2} \) |
| 7 | \( 1 + 5.74T + 49T^{2} \) |
| 11 | \( 1 + (-5.59 + 5.59i)T - 121iT^{2} \) |
| 13 | \( 1 + (13.5 - 13.5i)T - 169iT^{2} \) |
| 17 | \( 1 + 19.7T + 289T^{2} \) |
| 19 | \( 1 + (21.6 + 21.6i)T + 361iT^{2} \) |
| 23 | \( 1 + 24.9T + 529T^{2} \) |
| 29 | \( 1 + (1.50 - 1.50i)T - 841iT^{2} \) |
| 31 | \( 1 - 2.20iT - 961T^{2} \) |
| 37 | \( 1 + (-27.6 - 27.6i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-21.4 + 21.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 76.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-56.5 - 56.5i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-48.0 + 48.0i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (51.5 - 51.5i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-63.4 - 63.4i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 43.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 73.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 4.12iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (38.4 + 38.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 52.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 23.1T + 9.40e3T^{2} \) |
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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
−12.11683852784730572812734729945, −11.36806969529709063782543683367, −10.14638288945512874098201425869, −9.181010522231483353929967235851, −8.443031607755895994028158875497, −6.98708694077748555200481699386, −6.28739131984038355139740053337, −4.38231872036453389023391747881, −2.39451353829254491026770772904, −0.10825706632679258635360455875,
2.34602497499229229261589002136, 3.87236494851724605967228919680, 6.06100646258279122653551904143, 7.11190797096484576127138253823, 8.077811994296712131146298977132, 9.352091793159606088000350683017, 10.14396049609647729558315105515, 11.06326115498927314309107388635, 12.28698506021886136647534260942, 12.85583584980199403592705276027