| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·5-s + 2·6-s + 7-s + 4·8-s − 4·9-s + 4·10-s + 11-s + 3·12-s − 3·13-s + 2·14-s + 2·15-s + 5·16-s + 17-s − 8·18-s − 3·19-s + 6·20-s + 21-s + 2·22-s + 2·23-s + 4·24-s + 3·25-s − 6·26-s − 6·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s + 0.377·7-s + 1.41·8-s − 4/3·9-s + 1.26·10-s + 0.301·11-s + 0.866·12-s − 0.832·13-s + 0.534·14-s + 0.516·15-s + 5/4·16-s + 0.242·17-s − 1.88·18-s − 0.688·19-s + 1.34·20-s + 0.218·21-s + 0.426·22-s + 0.417·23-s + 0.816·24-s + 3/5·25-s − 1.17·26-s − 1.15·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.057957440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.057957440\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 63 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 113 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 27 T + 375 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.47883434393512256328344394257, −12.15772750434465927058119918655, −11.44686251142789410021465049921, −11.19558779221915460508966203868, −10.82084895697821165088256119833, −10.01325729367604094589154766938, −9.542030720090398116833551036626, −9.140306759551471116968745802587, −8.377340443911295124746456879666, −7.996795640306081590736648146812, −7.37284920887802757557061253147, −6.70407123397807318145608994971, −6.06854457591781683225897468344, −5.81865497788505764214963526118, −4.89069048655957089784278056400, −4.88440935172606354314966556919, −3.67058546540356043216825506916, −3.21835418156143581887086605491, −2.38964411962337051053951412739, −1.89317783414266679508333114377,
1.89317783414266679508333114377, 2.38964411962337051053951412739, 3.21835418156143581887086605491, 3.67058546540356043216825506916, 4.88440935172606354314966556919, 4.89069048655957089784278056400, 5.81865497788505764214963526118, 6.06854457591781683225897468344, 6.70407123397807318145608994971, 7.37284920887802757557061253147, 7.996795640306081590736648146812, 8.377340443911295124746456879666, 9.140306759551471116968745802587, 9.542030720090398116833551036626, 10.01325729367604094589154766938, 10.82084895697821165088256119833, 11.19558779221915460508966203868, 11.44686251142789410021465049921, 12.15772750434465927058119918655, 12.47883434393512256328344394257
