| L(s) = 1 | − 0.490·5-s − 2.64i·7-s + 15.5i·11-s + 3.50·13-s + 24.1·17-s − 3.56i·19-s − 19.5i·23-s − 24.7·25-s + 10.9·29-s − 21.1i·31-s + 1.29i·35-s + 58.4·37-s − 54.1·41-s + 35.6i·43-s + 64.2i·47-s + ⋯ |
| L(s) = 1 | − 0.0980·5-s − 0.377i·7-s + 1.41i·11-s + 0.269·13-s + 1.42·17-s − 0.187i·19-s − 0.851i·23-s − 0.990·25-s + 0.378·29-s − 0.683i·31-s + 0.0370i·35-s + 1.57·37-s − 1.32·41-s + 0.828i·43-s + 1.36i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.910567725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910567725\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 + 0.490T + 25T^{2} \) |
| 11 | \( 1 - 15.5iT - 121T^{2} \) |
| 13 | \( 1 - 3.50T + 169T^{2} \) |
| 17 | \( 1 - 24.1T + 289T^{2} \) |
| 19 | \( 1 + 3.56iT - 361T^{2} \) |
| 23 | \( 1 + 19.5iT - 529T^{2} \) |
| 29 | \( 1 - 10.9T + 841T^{2} \) |
| 31 | \( 1 + 21.1iT - 961T^{2} \) |
| 37 | \( 1 - 58.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 54.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 35.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 64.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 87.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 66.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 21.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 64.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 99.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 139. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.03iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 23.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 171.T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.266523429762214814925669766102, −7.896043435948364378991597494902, −7.79704385390719366101824184949, −6.69880918066722544542900442304, −5.98856006039891687649836798334, −4.87528416011358582591926399354, −4.25528840286264218911586931738, −3.22529659709668625789641438501, −2.10634140875638387355172344740, −0.957836417453114833973034344763,
0.59862510529094673392568028528, 1.79190029212925728936622272662, 3.20962907647488606088815228575, 3.61094040972310828653660618656, 4.99197920504844063985763217471, 5.74361803299832677426898788188, 6.30919927037706552000977394638, 7.46137547304902754515853400390, 8.164782288668232485565438195814, 8.730575956200259194714923867677
