| L(s) = 1 | − 2·3-s − 4·5-s + 4·7-s + 3·9-s − 4·11-s + 8·15-s − 4·17-s − 8·21-s + 8·23-s + 4·25-s − 4·27-s − 4·29-s + 12·31-s + 8·33-s − 16·35-s + 8·37-s − 12·41-s + 8·43-s − 12·45-s + 8·47-s + 8·51-s − 4·53-s + 16·55-s + 8·59-s + 8·61-s + 12·63-s + 16·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 1.51·7-s + 9-s − 1.20·11-s + 2.06·15-s − 0.970·17-s − 1.74·21-s + 1.66·23-s + 4/5·25-s − 0.769·27-s − 0.742·29-s + 2.15·31-s + 1.39·33-s − 2.70·35-s + 1.31·37-s − 1.87·41-s + 1.21·43-s − 1.78·45-s + 1.16·47-s + 1.12·51-s − 0.549·53-s + 2.15·55-s + 1.04·59-s + 1.02·61-s + 1.51·63-s + 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.123866691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123866691\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 96 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 240 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.720657465588992741323648898354, −9.307725627013187874313445401912, −8.462696968055348274953213024643, −8.453802432578219914454592935911, −7.987425745198345241223595938623, −7.72251651039203290486235916671, −7.31948487981926775570545861937, −6.91565713363923914392457449652, −6.44334854620729196704432493871, −6.03328648536811047901589859225, −5.15913269219330722514288688100, −5.11129558682569747101636526948, −4.74033360858085745345321212295, −4.41410227210658894190443191545, −3.68874342306983489244066282305, −3.49816416934195065424496586633, −2.35318901422880603439814620623, −2.21290179884846859514522399588, −0.920746276593866604600437958299, −0.61354939182090444361538852552,
0.61354939182090444361538852552, 0.920746276593866604600437958299, 2.21290179884846859514522399588, 2.35318901422880603439814620623, 3.49816416934195065424496586633, 3.68874342306983489244066282305, 4.41410227210658894190443191545, 4.74033360858085745345321212295, 5.11129558682569747101636526948, 5.15913269219330722514288688100, 6.03328648536811047901589859225, 6.44334854620729196704432493871, 6.91565713363923914392457449652, 7.31948487981926775570545861937, 7.72251651039203290486235916671, 7.987425745198345241223595938623, 8.453802432578219914454592935911, 8.462696968055348274953213024643, 9.307725627013187874313445401912, 9.720657465588992741323648898354
