| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 2·5-s + 4·6-s + 4·8-s − 9-s + 4·10-s − 4·11-s + 6·12-s + 4·13-s + 4·15-s + 5·16-s + 2·17-s − 2·18-s + 6·20-s − 8·22-s + 8·24-s + 25-s + 8·26-s − 6·27-s − 10·29-s + 8·30-s − 14·31-s + 6·32-s − 8·33-s + 4·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 1.41·8-s − 1/3·9-s + 1.26·10-s − 1.20·11-s + 1.73·12-s + 1.10·13-s + 1.03·15-s + 5/4·16-s + 0.485·17-s − 0.471·18-s + 1.34·20-s − 1.70·22-s + 1.63·24-s + 1/5·25-s + 1.56·26-s − 1.15·27-s − 1.85·29-s + 1.46·30-s − 2.51·31-s + 1.06·32-s − 1.39·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.70327196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.70327196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 14 T + 109 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 16 T + 136 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 77 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 112 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 20 T + 226 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 165 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 248 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 171 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 195 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.644907930985845582965819274730, −8.149498604435664574831816036032, −7.72942534799284651813487818808, −7.66442045379862573925047676400, −7.12908054837294707445034687046, −6.86123036651845838446535259478, −6.01327976419981801911694296325, −5.87159748867134372785048940290, −5.74869129574092192796578771888, −5.45895702909368595861200797339, −4.86475501906090255890633155631, −4.46304335321314618777407361582, −3.76093908948347490920581515469, −3.73378178317994568631739624083, −3.10513150521653224350552901401, −2.92310639294324702810369196449, −2.31304907826566716748110824470, −1.99748729167342234837029981045, −1.64982416714107167515857975677, −0.64326099353693652044440518949,
0.64326099353693652044440518949, 1.64982416714107167515857975677, 1.99748729167342234837029981045, 2.31304907826566716748110824470, 2.92310639294324702810369196449, 3.10513150521653224350552901401, 3.73378178317994568631739624083, 3.76093908948347490920581515469, 4.46304335321314618777407361582, 4.86475501906090255890633155631, 5.45895702909368595861200797339, 5.74869129574092192796578771888, 5.87159748867134372785048940290, 6.01327976419981801911694296325, 6.86123036651845838446535259478, 7.12908054837294707445034687046, 7.66442045379862573925047676400, 7.72942534799284651813487818808, 8.149498604435664574831816036032, 8.644907930985845582965819274730
