L(s) = 1 | − 3i·3-s − 10.3i·5-s − 3.46·7-s − 9·9-s − 55.4i·13-s − 31.1·15-s − 90·17-s + 116i·19-s + 10.3i·21-s − 103.·23-s + 17·25-s + 27i·27-s − 259. i·29-s − 301.·31-s + 36i·35-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.929i·5-s − 0.187·7-s − 0.333·9-s − 1.18i·13-s − 0.536·15-s − 1.28·17-s + 1.40i·19-s + 0.107i·21-s − 0.942·23-s + 0.136·25-s + 0.192i·27-s − 1.66i·29-s − 1.74·31-s + 0.173i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.120766 - 0.917310i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.120766 - 0.917310i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
good | 5 | \( 1 + 10.3iT - 125T^{2} \) |
| 7 | \( 1 + 3.46T + 343T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + 55.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 90T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 259. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 301.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 34.6iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 54T + 6.89e4T^{2} \) |
| 43 | \( 1 + 20iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 488. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 324iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 575. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 116iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 148.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 918T + 7.04e5T^{2} \) |
| 97 | \( 1 - 190T + 9.12e5T^{2} \) |
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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
−11.86666870918728871999911308416, −10.68040472437362072982510926194, −9.549438467199066509943249325939, −8.444558517412407775253868476438, −7.71557812046204282866431922030, −6.28404861270782720116298964256, −5.30427207067229797905608829478, −3.83949570730201878639250192108, −2.03181003938150826439498077059, −0.38459347818424065981239101587,
2.32070273420525050174313754597, 3.70082701931104018595896119667, 4.91639952454929353037505620538, 6.46874386512334420223438416424, 7.16050717905370848945451251612, 8.811460502961056568587749967068, 9.482338789018893215748026467942, 10.87964217928275438876686715944, 11.10969092885400219690465149831, 12.46453461442351283062180575682