| L(s) = 1 | − 4-s + 5-s − 8·7-s − 2·8-s + 9-s + 4·11-s + 3·13-s + 16-s + 8·17-s − 4·19-s − 20-s − 2·25-s + 8·28-s − 16·29-s + 12·31-s + 4·32-s − 8·35-s − 36-s − 2·40-s − 16·43-s − 4·44-s + 45-s + 34·49-s − 3·52-s + 4·55-s + 16·56-s + 12·59-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.447·5-s − 3.02·7-s − 0.707·8-s + 1/3·9-s + 1.20·11-s + 0.832·13-s + 1/4·16-s + 1.94·17-s − 0.917·19-s − 0.223·20-s − 2/5·25-s + 1.51·28-s − 2.97·29-s + 2.15·31-s + 0.707·32-s − 1.35·35-s − 1/6·36-s − 0.316·40-s − 2.43·43-s − 0.603·44-s + 0.149·45-s + 34/7·49-s − 0.416·52-s + 0.539·55-s + 2.13·56-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4618358169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4618358169\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−19.4604421777, −19.1626037497, −18.6983747433, −18.4282714936, −17.2849148009, −16.9011141517, −16.4135833599, −16.0225385141, −14.9691010860, −14.9380436528, −13.6745032861, −13.3351916393, −12.9568449998, −12.1436052702, −11.8538765065, −10.3916050944, −9.95280234664, −9.30558712287, −9.19274055467, −7.94123132226, −6.54556545043, −6.42217617653, −5.65486780280, −3.71205447024, −3.36585804146,
3.36585804146, 3.71205447024, 5.65486780280, 6.42217617653, 6.54556545043, 7.94123132226, 9.19274055467, 9.30558712287, 9.95280234664, 10.3916050944, 11.8538765065, 12.1436052702, 12.9568449998, 13.3351916393, 13.6745032861, 14.9380436528, 14.9691010860, 16.0225385141, 16.4135833599, 16.9011141517, 17.2849148009, 18.4282714936, 18.6983747433, 19.1626037497, 19.4604421777
