| L(s) = 1 | − 2.23i·5-s − 10.9i·7-s + 15.6i·11-s + 4.14·13-s + 22.0i·17-s − 9.79i·19-s + (−6.20 − 22.1i)23-s − 5.00·25-s + 46.1·29-s + 22.4·31-s − 24.4·35-s + 63.2i·37-s − 48.0·41-s − 26.9i·43-s + 42.7·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 1.56i·7-s + 1.42i·11-s + 0.318·13-s + 1.29i·17-s − 0.515i·19-s + (−0.269 − 0.962i)23-s − 0.200·25-s + 1.59·29-s + 0.725·31-s − 0.699·35-s + 1.70i·37-s − 1.17·41-s − 0.625i·43-s + 0.909·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.969028800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.969028800\) |
| \(L(2)\) |
|
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 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (6.20 + 22.1i)T \) |
| good | 7 | \( 1 + 10.9iT - 49T^{2} \) |
| 11 | \( 1 - 15.6iT - 121T^{2} \) |
| 13 | \( 1 - 4.14T + 169T^{2} \) |
| 17 | \( 1 - 22.0iT - 289T^{2} \) |
| 19 | \( 1 + 9.79iT - 361T^{2} \) |
| 29 | \( 1 - 46.1T + 841T^{2} \) |
| 31 | \( 1 - 22.4T + 961T^{2} \) |
| 37 | \( 1 - 63.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 48.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 26.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 42.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 48.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 11.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 56.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 83.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 29.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 134.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 100. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 2.18iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 67.6iT - 9.40e3T^{2} \) |
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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
−8.250893676979201672913451721520, −7.56347220818220259809590945990, −6.74609952419425989133117485574, −6.37157074813487332627269979913, −5.04560252471671064139366594455, −4.39445075528551607074966651334, −3.99756666474918606739548065127, −2.78655462065142115420699782053, −1.60290998465897820725495675452, −0.843811790403885789545342780877,
0.51510088064397334724093964703, 1.85984965630419291941840911821, 2.90248803938288256000414587387, 3.24556519404300116551001411387, 4.50173484596766645012366217807, 5.57996756849871389434030332517, 5.80425739800496265759920907535, 6.63912380226812156481255795114, 7.53344433782665644444591009780, 8.440809342780904167854831936930
