| L(s) = 1 | + 9·5-s − 36·9-s + 6·11-s − 76·19-s + 25·25-s − 324·45-s + 73·49-s + 54·55-s + 206·61-s + 810·81-s − 684·95-s − 216·99-s + 408·101-s + 251·121-s − 54·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 2.73e3·171-s + 173-s + ⋯ |
| L(s) = 1 | + 9/5·5-s − 4·9-s + 6/11·11-s − 4·19-s + 25-s − 7.19·45-s + 1.48·49-s + 0.981·55-s + 3.37·61-s + 10·81-s − 7.19·95-s − 2.18·99-s + 4.03·101-s + 2.07·121-s − 0.431·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4·169-s + 16·171-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.873678481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.873678481\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 9 T + 56 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 5 T - 24 T^{2} - 5 p^{2} T^{3} + p^{4} T^{4} )( 1 + 5 T - 24 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 112 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 15 T - 64 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} )( 1 + 15 T - 64 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2}( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 85 T + 5376 T^{2} - 85 p^{2} T^{3} + p^{4} T^{4} )( 1 + 85 T + 5376 T^{2} + 85 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 75 T + 3416 T^{2} - 75 p^{2} T^{3} + p^{4} T^{4} )( 1 + 75 T + 3416 T^{2} + 75 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 103 T + 6888 T^{2} - 103 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 25 T - 4704 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} )( 1 + 25 T - 4704 T^{2} + 25 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )^{2}( 1 + 90 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
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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
−8.030056034697041029272178998588, −8.012102840132713892493159068454, −7.59328391228926338863015565025, −7.03417414972920917516311102155, −6.99912113461432831028506833672, −6.45097818617978536355178950109, −6.41169093787028445038474743640, −6.04440355574004602385604384382, −5.98869635432408076563673816301, −5.78876797994101388213218384341, −5.75663178723311824835501360869, −4.98842543503486023617978898820, −4.98019738760713183467793230690, −4.94724755994227988532532015744, −4.14458021363873305072041519107, −3.93284131387965105148104834508, −3.62791571220685113820423010922, −3.30417424409565413749275365599, −2.74443997547090919224772920435, −2.43818391686994095334391148065, −2.33284381781232185483145750099, −2.12907155089320428550870309967, −1.70731235583498825174625782280, −0.64418033744644635485087214622, −0.38703907274683668452437268089,
0.38703907274683668452437268089, 0.64418033744644635485087214622, 1.70731235583498825174625782280, 2.12907155089320428550870309967, 2.33284381781232185483145750099, 2.43818391686994095334391148065, 2.74443997547090919224772920435, 3.30417424409565413749275365599, 3.62791571220685113820423010922, 3.93284131387965105148104834508, 4.14458021363873305072041519107, 4.94724755994227988532532015744, 4.98019738760713183467793230690, 4.98842543503486023617978898820, 5.75663178723311824835501360869, 5.78876797994101388213218384341, 5.98869635432408076563673816301, 6.04440355574004602385604384382, 6.41169093787028445038474743640, 6.45097818617978536355178950109, 6.99912113461432831028506833672, 7.03417414972920917516311102155, 7.59328391228926338863015565025, 8.012102840132713892493159068454, 8.030056034697041029272178998588
