L(s) = 1 | − 5·7-s + 18·9-s − 3·11-s + 15·17-s − 38·19-s + 60·23-s − 85·43-s + 75·47-s + 49·49-s − 103·61-s − 90·63-s − 25·73-s + 15·77-s + 243·81-s − 180·83-s − 54·99-s − 204·101-s − 75·119-s + 121·121-s + 127-s + 131-s + 190·133-s + 137-s + 139-s + 149-s + 151-s + 270·153-s + ⋯ |
L(s) = 1 | − 5/7·7-s + 2·9-s − 0.272·11-s + 0.882·17-s − 2·19-s + 2.60·23-s − 1.97·43-s + 1.59·47-s + 49-s − 1.68·61-s − 1.42·63-s − 0.342·73-s + 0.194·77-s + 3·81-s − 2.16·83-s − 0.545·99-s − 2.01·101-s − 0.630·119-s + 121-s + 0.00787·127-s + 0.00763·131-s + 10/7·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 1.76·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.813152038\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.813152038\) |
\(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 5 T - 24 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 112 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 15 T - 64 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 85 T + 5376 T^{2} + 85 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 75 T + 3416 T^{2} - 75 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 103 T + 6888 T^{2} + 103 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 25 T - 4704 T^{2} + 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 90 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−9.144666601389194797669518280595, −9.010336556512219187556098014729, −8.313038279620157081545421914458, −8.237067162843546821653440422386, −7.45422462746594013909818768646, −7.18101921575324547552914729216, −6.82191021843979775329222725351, −6.78345162429324655950345296517, −5.93421300898144336301304374404, −5.81465927517559651812578099604, −4.95948762092247983190963069769, −4.82409167807978054393238039799, −4.19422490477833243840068315924, −4.02024930766860252786133704575, −3.22455640702439263994379769461, −3.01080058983811147757540988983, −2.27491851266803815172964022623, −1.63934699019100711331845484937, −1.17391223682618638231420277467, −0.46845207038472105041304254503,
0.46845207038472105041304254503, 1.17391223682618638231420277467, 1.63934699019100711331845484937, 2.27491851266803815172964022623, 3.01080058983811147757540988983, 3.22455640702439263994379769461, 4.02024930766860252786133704575, 4.19422490477833243840068315924, 4.82409167807978054393238039799, 4.95948762092247983190963069769, 5.81465927517559651812578099604, 5.93421300898144336301304374404, 6.78345162429324655950345296517, 6.82191021843979775329222725351, 7.18101921575324547552914729216, 7.45422462746594013909818768646, 8.237067162843546821653440422386, 8.313038279620157081545421914458, 9.010336556512219187556098014729, 9.144666601389194797669518280595