| L(s) = 1 | − 4·13-s + 4·37-s − 4·73-s − 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 4·13-s + 4·37-s − 4·73-s − 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4579882540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4579882540\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.41054305811444356417102486091, −6.15756211351203787298141746893, −6.11996903532565470714220458433, −5.71841996654793039583486992997, −5.59754026359606987233209693431, −5.25573181463499213637596401134, −5.14497657420040012058525392844, −5.11986727781763094305097409929, −4.78893589283817434425890294574, −4.52546195238292836567902063255, −4.23527474659811178759394437644, −4.22920663736593155309763152055, −4.04839360004186556122114587147, −3.95075396411524422332402542777, −3.15546935633479616241628653306, −3.11611641826183295277046956250, −2.93447964331481709106596383525, −2.63691476428635955911080118077, −2.55133007551641436562008994529, −2.28726575829011616403461018623, −2.15744213240418502592389045467, −1.55348687072724924556698143088, −1.38687071334865829826438095344, −1.00487331015154251525109507367, −0.29006347170982799871744771452,
0.29006347170982799871744771452, 1.00487331015154251525109507367, 1.38687071334865829826438095344, 1.55348687072724924556698143088, 2.15744213240418502592389045467, 2.28726575829011616403461018623, 2.55133007551641436562008994529, 2.63691476428635955911080118077, 2.93447964331481709106596383525, 3.11611641826183295277046956250, 3.15546935633479616241628653306, 3.95075396411524422332402542777, 4.04839360004186556122114587147, 4.22920663736593155309763152055, 4.23527474659811178759394437644, 4.52546195238292836567902063255, 4.78893589283817434425890294574, 5.11986727781763094305097409929, 5.14497657420040012058525392844, 5.25573181463499213637596401134, 5.59754026359606987233209693431, 5.71841996654793039583486992997, 6.11996903532565470714220458433, 6.15756211351203787298141746893, 6.41054305811444356417102486091
