| L(s) = 1 | − 2-s − 3-s − 4-s − 2·5-s + 6-s − 4·7-s + 3·8-s + 9-s + 2·10-s + 12-s + 2·13-s + 4·14-s + 2·15-s − 16-s + 2·17-s − 18-s + 2·20-s + 4·21-s + 8·23-s − 3·24-s − 25-s − 2·26-s − 27-s + 4·28-s + 6·29-s − 2·30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.447·20-s + 0.872·21-s + 1.66·23-s − 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s − 0.365·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4137690469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4137690469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−11.21317067856663011790716323208, −10.45490550189802125561108883297, −9.528487026476181059669700861222, −8.829662012954888204160857557729, −7.67376126887882242787787369865, −6.85553577597705614902881023761, −5.66464594246847012421851082372, −4.33065348032451946577135661956, −3.31119773648628104035349562784, −0.71891279628917143719551815370,
0.71891279628917143719551815370, 3.31119773648628104035349562784, 4.33065348032451946577135661956, 5.66464594246847012421851082372, 6.85553577597705614902881023761, 7.67376126887882242787787369865, 8.829662012954888204160857557729, 9.528487026476181059669700861222, 10.45490550189802125561108883297, 11.21317067856663011790716323208
