L(s) = 1 | + 2.23i·5-s + 5.48·7-s + 9.17i·11-s − 11.4·13-s − 16.9i·17-s − 26.9·19-s − 4.93i·23-s − 5.00·25-s − 20.5i·29-s + 20.9·31-s + 12.2i·35-s + 62.4·37-s + 40.9i·41-s − 1.02·43-s − 86.2i·47-s + ⋯ |
L(s) = 1 | + 0.447i·5-s + 0.783·7-s + 0.833i·11-s − 0.883·13-s − 0.998i·17-s − 1.41·19-s − 0.214i·23-s − 0.200·25-s − 0.707i·29-s + 0.676·31-s + 0.350i·35-s + 1.68·37-s + 0.998i·41-s − 0.0238·43-s − 1.83i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.814856095\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.814856095\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 - 5.48T + 49T^{2} \) |
| 11 | \( 1 - 9.17iT - 121T^{2} \) |
| 13 | \( 1 + 11.4T + 169T^{2} \) |
| 17 | \( 1 + 16.9iT - 289T^{2} \) |
| 19 | \( 1 + 26.9T + 361T^{2} \) |
| 23 | \( 1 + 4.93iT - 529T^{2} \) |
| 29 | \( 1 + 20.5iT - 841T^{2} \) |
| 31 | \( 1 - 20.9T + 961T^{2} \) |
| 37 | \( 1 - 62.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 40.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.02T + 1.84e3T^{2} \) |
| 47 | \( 1 + 86.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 96.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 112. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 66.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76T + 4.48e3T^{2} \) |
| 71 | \( 1 - 24.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 18.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 106.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 45.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 87.0T + 9.40e3T^{2} \) |
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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
−8.208030400033248651462176417849, −7.965534448331893249430767227863, −6.87300212841212382820396410463, −6.52054456900075295461086773326, −5.20690837318545624164673372360, −4.71662452877760202553494275009, −3.86216652590826416103297660016, −2.52074632149890820205208226825, −2.05427626253472925262526158507, −0.49749027980153699482417549770,
0.882258569866188961204681124014, 1.93544225375136720590782335511, 2.91383745913271619718211748796, 4.17916768644048147655878025408, 4.62020967541014593362133648943, 5.68204346820005332631690490509, 6.21881703075578863977763578287, 7.30072931656209547182058206189, 8.061271696203809131723285509839, 8.563707825839943065182584714629