| L(s) = 1 | − i·2-s + 3-s − 4-s − i·6-s + i·8-s + 9-s − 4i·11-s − 12-s + (3 + 2i)13-s + 16-s − 4·17-s − i·18-s − 4i·19-s − 4·22-s − 6·23-s + i·24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.408i·6-s + 0.353i·8-s + 0.333·9-s − 1.20i·11-s − 0.288·12-s + (0.832 + 0.554i)13-s + 0.250·16-s − 0.970·17-s − 0.235i·18-s − 0.917i·19-s − 0.852·22-s − 1.25·23-s + 0.204i·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.765804969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.765804969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3 - 2i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
| 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
−9.019722307439620712715969406652, −8.363539602907481682457783862403, −7.59378688054193530172664770070, −6.41601366300404473739365371090, −5.78747342078323616494685219749, −4.42704391260931184504391954888, −3.90779067260964431617456351828, −2.84803387063646909688736294825, −2.02271107803511629709079751488, −0.61343835805002563251832578519,
1.43611284685129411138177216473, 2.64932643699283655235127342336, 3.87800635582167315382578870137, 4.48101267794303323257163145678, 5.55735063006930008333610860599, 6.39445224184079194766537324328, 7.17256180147514723477012595084, 7.891830857093789402327855086608, 8.608139741057166439950524631461, 9.187613576960461566352025134747
