L(s) = 1 | + (−1.37 + 1.45i)2-s − 5.29i·3-s + (−0.216 − 3.99i)4-s − 4.75·5-s + (7.69 + 7.28i)6-s − 1.34i·7-s + (6.09 + 5.18i)8-s − 19.0·9-s + (6.53 − 6.90i)10-s + 16.3i·11-s + (−21.1 + 1.14i)12-s − 13.1·13-s + (1.95 + 1.85i)14-s + 25.1i·15-s + (−15.9 + 1.72i)16-s + 5.86·17-s + ⋯ |
L(s) = 1 | + (−0.687 + 0.725i)2-s − 1.76i·3-s + (−0.0540 − 0.998i)4-s − 0.950·5-s + (1.28 + 1.21i)6-s − 0.192i·7-s + (0.762 + 0.647i)8-s − 2.11·9-s + (0.653 − 0.690i)10-s + 1.48i·11-s + (−1.76 + 0.0954i)12-s − 1.01·13-s + (0.139 + 0.132i)14-s + 1.67i·15-s + (−0.994 + 0.107i)16-s + 0.345·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0540i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00812626 - 0.300647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00812626 - 0.300647i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.37 - 1.45i)T \) |
| 31 | \( 1 - 5.56iT \) |
good | 3 | \( 1 + 5.29iT - 9T^{2} \) |
| 5 | \( 1 + 4.75T + 25T^{2} \) |
| 7 | \( 1 + 1.34iT - 49T^{2} \) |
| 11 | \( 1 - 16.3iT - 121T^{2} \) |
| 13 | \( 1 + 13.1T + 169T^{2} \) |
| 17 | \( 1 - 5.86T + 289T^{2} \) |
| 19 | \( 1 + 17.9iT - 361T^{2} \) |
| 23 | \( 1 + 33.6iT - 529T^{2} \) |
| 29 | \( 1 + 15.6T + 841T^{2} \) |
| 37 | \( 1 + 35.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 32.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 70.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 68.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 19.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 57.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 115.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 85.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 23.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 101.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 65.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 10.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 66.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 168.T + 9.40e3T^{2} \) |
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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
−12.50408659455144062604900198891, −11.95131359150077201665386909749, −10.56969607343525610440931353941, −9.087667791135409758609470625054, −7.85191820007209836163866525356, −7.32247722677633495553562304884, −6.60008339457408828329561361682, −4.88856733207450819699856171581, −2.10695384156154525492913293290, −0.25114535690379505774372507245,
3.22447907635202585478736516848, 3.95305934313305976614365218729, 5.43539436342691533780488393624, 7.74183078368691815129333313766, 8.681841445947284814300376666987, 9.639836436425985982715170019874, 10.49528307984663912993702276920, 11.44246753646045875816701091098, 11.99171786375205095887825771956, 13.71891618080587683097003905047