| L(s) = 1 | + 2-s + 2·4-s − 5-s + 3·7-s + 5·8-s − 10-s − 2·11-s + 2·13-s + 3·14-s + 5·16-s − 8·17-s − 16·19-s − 2·20-s − 2·22-s + 3·23-s + 2·26-s + 6·28-s − 29-s + 10·32-s − 8·34-s − 3·35-s − 8·37-s − 16·38-s − 5·40-s + 5·41-s + 8·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 0.447·5-s + 1.13·7-s + 1.76·8-s − 0.316·10-s − 0.603·11-s + 0.554·13-s + 0.801·14-s + 5/4·16-s − 1.94·17-s − 3.67·19-s − 0.447·20-s − 0.426·22-s + 0.625·23-s + 0.392·26-s + 1.13·28-s − 0.185·29-s + 1.76·32-s − 1.37·34-s − 0.507·35-s − 1.31·37-s − 2.59·38-s − 0.790·40-s + 0.780·41-s + 1.21·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.910058152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910058152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−13.53536974688294486531507335638, −13.06500413034760913099630159867, −12.44720736127108332222042886792, −12.14553749369660095056465988257, −11.06415228805254090880947695555, −11.05477000469114359345954208991, −10.65103097021576409407624710988, −10.51702499842888202310627889966, −8.921841941855148806734634355528, −8.855252413022779317771711752149, −7.957875702179757525120975719397, −7.75429630304440734716981855196, −6.76866160816444114731230997675, −6.58473244006360102269879629692, −5.73972285476685472650438434219, −4.69217308439735030069943619664, −4.47984723041382379716095510736, −3.92453812925775570334282128180, −2.35354807392313807706068771669, −1.98693423884418312478743050279,
1.98693423884418312478743050279, 2.35354807392313807706068771669, 3.92453812925775570334282128180, 4.47984723041382379716095510736, 4.69217308439735030069943619664, 5.73972285476685472650438434219, 6.58473244006360102269879629692, 6.76866160816444114731230997675, 7.75429630304440734716981855196, 7.957875702179757525120975719397, 8.855252413022779317771711752149, 8.921841941855148806734634355528, 10.51702499842888202310627889966, 10.65103097021576409407624710988, 11.05477000469114359345954208991, 11.06415228805254090880947695555, 12.14553749369660095056465988257, 12.44720736127108332222042886792, 13.06500413034760913099630159867, 13.53536974688294486531507335638
