| L(s) = 1 | − 4·3-s − 4·5-s − 4·7-s + 8·9-s − 2·13-s + 16·15-s − 10·17-s − 8·19-s + 16·21-s − 4·23-s + 11·25-s − 12·27-s + 16·35-s + 2·37-s + 8·39-s + 12·43-s − 32·45-s + 4·47-s + 8·49-s + 40·51-s − 14·53-s + 32·57-s − 8·59-s − 8·61-s − 32·63-s + 8·65-s + 20·67-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.78·5-s − 1.51·7-s + 8/3·9-s − 0.554·13-s + 4.13·15-s − 2.42·17-s − 1.83·19-s + 3.49·21-s − 0.834·23-s + 11/5·25-s − 2.30·27-s + 2.70·35-s + 0.328·37-s + 1.28·39-s + 1.82·43-s − 4.77·45-s + 0.583·47-s + 8/7·49-s + 5.60·51-s − 1.92·53-s + 4.23·57-s − 1.04·59-s − 1.02·61-s − 4.03·63-s + 0.992·65-s + 2.44·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−12.39120829607980716833016442920, −12.24639074133331883143606070816, −11.53149773793812008891315253886, −11.21213632369939348194190299961, −10.72216362126657541072761406516, −10.65330872072270715417064095239, −9.694235667765133118993808626927, −9.155775512676237659889010969563, −8.493258516651058848658419487236, −7.79991666445786904202477652183, −6.98204015859890003664557088210, −6.76496165610320624134231613644, −6.17238974846732228512913187291, −5.82427128957150840200781620625, −4.62121038175294800783863441427, −4.44760348718601702809622473482, −3.81985278896428350058783924405, −2.55316889778586010739924050045, 0, 0,
2.55316889778586010739924050045, 3.81985278896428350058783924405, 4.44760348718601702809622473482, 4.62121038175294800783863441427, 5.82427128957150840200781620625, 6.17238974846732228512913187291, 6.76496165610320624134231613644, 6.98204015859890003664557088210, 7.79991666445786904202477652183, 8.493258516651058848658419487236, 9.155775512676237659889010969563, 9.694235667765133118993808626927, 10.65330872072270715417064095239, 10.72216362126657541072761406516, 11.21213632369939348194190299961, 11.53149773793812008891315253886, 12.24639074133331883143606070816, 12.39120829607980716833016442920
