| L(s) = 1 | + 2-s − 4-s + 2·5-s + 7-s − 3·8-s + 2·10-s + 2·13-s + 14-s − 16-s − 2·17-s − 6·19-s − 2·20-s + 4·23-s − 25-s + 2·26-s − 28-s + 29-s + 4·31-s + 5·32-s − 2·34-s + 2·35-s + 6·37-s − 6·38-s − 6·40-s − 2·41-s + 12·43-s + 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s − 1.06·8-s + 0.632·10-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.447·20-s + 0.834·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.185·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s + 0.338·35-s + 0.986·37-s − 0.973·38-s − 0.948·40-s − 0.312·41-s + 1.82·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.327751863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.327751863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.04365035231443, −12.74347934918093, −12.10294905418038, −11.71096221916270, −11.09791566791193, −10.60235622898172, −10.29939951712359, −9.590861440919524, −9.151543528874530, −8.830878415459815, −8.388853976868034, −7.764664385391316, −7.220627461234723, −6.377672599822713, −6.113955021352733, −5.900824645890685, −5.064029807787423, −4.682194409016222, −4.287180932853477, −3.734199296583972, −2.993379986000142, −2.533511552236624, −1.915190399907115, −1.214591606934073, −0.4650078083053110,
0.4650078083053110, 1.214591606934073, 1.915190399907115, 2.533511552236624, 2.993379986000142, 3.734199296583972, 4.287180932853477, 4.682194409016222, 5.064029807787423, 5.900824645890685, 6.113955021352733, 6.377672599822713, 7.220627461234723, 7.764664385391316, 8.388853976868034, 8.830878415459815, 9.151543528874530, 9.590861440919524, 10.29939951712359, 10.60235622898172, 11.09791566791193, 11.71096221916270, 12.10294905418038, 12.74347934918093, 13.04365035231443
