| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 11-s + 2·13-s + 16-s + 2·17-s + 8·19-s + 20-s − 22-s − 4·23-s + 25-s + 2·26-s − 2·29-s + 8·31-s + 32-s + 2·34-s − 2·37-s + 8·38-s + 40-s − 6·41-s + 8·43-s − 44-s − 4·46-s + 4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 1.83·19-s + 0.223·20-s − 0.213·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s − 0.328·37-s + 1.29·38-s + 0.158·40-s − 0.937·41-s + 1.21·43-s − 0.150·44-s − 0.589·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.793052956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.793052956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.04005729684025589920837492028, −9.320715839340683774661507050378, −8.145914445140422560558086809244, −7.42915492399457791026016616809, −6.36062597601729567707034918033, −5.63902222814118228132870695546, −4.83318849190237929422060153274, −3.64982117311005000207662650100, −2.74212216031092786291276097908, −1.35568641827293212965393981435,
1.35568641827293212965393981435, 2.74212216031092786291276097908, 3.64982117311005000207662650100, 4.83318849190237929422060153274, 5.63902222814118228132870695546, 6.36062597601729567707034918033, 7.42915492399457791026016616809, 8.145914445140422560558086809244, 9.320715839340683774661507050378, 10.04005729684025589920837492028
