| L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 5.79i·5-s − 8.39·7-s − 2.82i·8-s + 8.19·10-s − 14.6i·11-s − 21.1·13-s − 11.8i·14-s + 4.00·16-s − 7.76i·17-s + 24.3·19-s + 11.5i·20-s + 20.7·22-s + 14.6i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.15i·5-s − 1.19·7-s − 0.353i·8-s + 0.819·10-s − 1.33i·11-s − 1.63·13-s − 0.847i·14-s + 0.250·16-s − 0.456i·17-s + 1.28·19-s + 0.579i·20-s + 0.944·22-s + 0.638i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.521183 - 0.521183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.521183 - 0.521183i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.79iT - 25T^{2} \) |
| 7 | \( 1 + 8.39T + 49T^{2} \) |
| 11 | \( 1 + 14.6iT - 121T^{2} \) |
| 13 | \( 1 + 21.1T + 169T^{2} \) |
| 17 | \( 1 + 7.76iT - 289T^{2} \) |
| 19 | \( 1 - 24.3T + 361T^{2} \) |
| 23 | \( 1 - 14.6iT - 529T^{2} \) |
| 29 | \( 1 + 35.4iT - 841T^{2} \) |
| 31 | \( 1 - 8T + 961T^{2} \) |
| 37 | \( 1 + 60.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 9.17T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 25.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 61.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 13T + 3.72e3T^{2} \) |
| 67 | \( 1 - 21.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 101. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 98.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 103. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 134. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 75.1T + 9.40e3T^{2} \) |
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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
−12.55651257118584117011272674573, −11.66388444170732229348921375105, −9.861926113191664640397392905371, −9.337243262165827220266159721401, −8.218352165724532609751907934433, −7.09921080175870541973625491972, −5.80429959425865506922202967372, −4.89008171240611499971508953300, −3.24597550965459115830460010404, −0.44371823779456661250118975195,
2.39784246845985934221107743062, 3.44886686862399144321536145439, 5.04955060995645206522782137467, 6.74574591659903942178735779389, 7.39830559953312551651555052357, 9.264583324407458929940317232518, 10.04269060926471521898508674040, 10.62327279923621066821978052404, 12.18089270372068185338604913479, 12.46755384587486142158891540406
