| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 5·16-s − 4·17-s − 25-s − 4·31-s − 6·32-s + 8·34-s − 2·49-s + 2·50-s + 8·62-s + 7·64-s − 12·68-s + 4·98-s − 3·100-s − 121-s − 12·124-s + 127-s − 8·128-s + 131-s + 16·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 5·16-s − 4·17-s − 25-s − 4·31-s − 6·32-s + 8·34-s − 2·49-s + 2·50-s + 8·62-s + 7·64-s − 12·68-s + 4·98-s − 3·100-s − 121-s − 12·124-s + 127-s − 8·128-s + 131-s + 16·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05497860248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05497860248\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.004775370035875678549801517474, −8.509323935427188817897226767673, −8.209330854520343721483777713500, −7.82798336254782488132942299808, −7.37504485776600740668160493186, −6.99065378156788487976903773723, −6.90769472325328276005129820196, −6.40850687194899238472655216858, −6.16880070300042160950176763394, −5.61768290524021930345307389743, −5.32106622713297737144981824007, −4.61785614930289404693837308299, −4.24219269957321971810072800774, −3.49016413452305190273510091926, −3.47720801558350251072351154824, −2.38616257920987093853028129461, −2.37150468565054318804073876501, −1.81169405165641799571091610747, −1.53852408418074802007929967237, −0.16795327581745304143397884528,
0.16795327581745304143397884528, 1.53852408418074802007929967237, 1.81169405165641799571091610747, 2.37150468565054318804073876501, 2.38616257920987093853028129461, 3.47720801558350251072351154824, 3.49016413452305190273510091926, 4.24219269957321971810072800774, 4.61785614930289404693837308299, 5.32106622713297737144981824007, 5.61768290524021930345307389743, 6.16880070300042160950176763394, 6.40850687194899238472655216858, 6.90769472325328276005129820196, 6.99065378156788487976903773723, 7.37504485776600740668160493186, 7.82798336254782488132942299808, 8.209330854520343721483777713500, 8.509323935427188817897226767673, 9.004775370035875678549801517474
