| L(s) = 1 | − 2·2-s + 4-s − 4·7-s + 2·8-s + 9-s − 4·11-s + 8·14-s − 3·16-s + 8·17-s − 2·18-s + 8·22-s + 25-s − 4·28-s − 8·29-s + 8·31-s − 2·32-s − 16·34-s + 36-s − 8·37-s + 4·41-s − 4·44-s + 8·47-s + 2·49-s − 2·50-s − 16·53-s − 8·56-s + 16·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.51·7-s + 0.707·8-s + 1/3·9-s − 1.20·11-s + 2.13·14-s − 3/4·16-s + 1.94·17-s − 0.471·18-s + 1.70·22-s + 1/5·25-s − 0.755·28-s − 1.48·29-s + 1.43·31-s − 0.353·32-s − 2.74·34-s + 1/6·36-s − 1.31·37-s + 0.624·41-s − 0.603·44-s + 1.16·47-s + 2/7·49-s − 0.282·50-s − 2.19·53-s − 1.06·56-s + 2.10·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1956145387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1956145387\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−19.1626037497, −19.1104624590, −18.6983747433, −18.1058537571, −17.2359534775, −16.9011141517, −16.0225385141, −15.9590260302, −14.9691010860, −14.0069258505, −13.3351916393, −12.6461787614, −12.1436052702, −10.6789224512, −10.3916050944, −9.52345167581, −9.30558712287, −7.94123132226, −7.66488013442, −6.42217617653, −5.23920392625, −3.36585804146,
3.36585804146, 5.23920392625, 6.42217617653, 7.66488013442, 7.94123132226, 9.30558712287, 9.52345167581, 10.3916050944, 10.6789224512, 12.1436052702, 12.6461787614, 13.3351916393, 14.0069258505, 14.9691010860, 15.9590260302, 16.0225385141, 16.9011141517, 17.2359534775, 18.1058537571, 18.6983747433, 19.1104624590, 19.1626037497
