L(s) = 1 | − 2·5-s − 7-s − 2·11-s − 6·13-s + 8·17-s − 19-s − 8·23-s + 5·25-s − 8·29-s + 3·31-s + 2·35-s + 37-s − 12·41-s − 22·43-s − 6·47-s − 6·49-s − 12·53-s + 4·55-s − 4·59-s + 6·61-s + 12·65-s + 13·67-s − 20·71-s + 11·73-s + 2·77-s − 3·79-s + 4·83-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.377·7-s − 0.603·11-s − 1.66·13-s + 1.94·17-s − 0.229·19-s − 1.66·23-s + 25-s − 1.48·29-s + 0.538·31-s + 0.338·35-s + 0.164·37-s − 1.87·41-s − 3.35·43-s − 0.875·47-s − 6/7·49-s − 1.64·53-s + 0.539·55-s − 0.520·59-s + 0.768·61-s + 1.48·65-s + 1.58·67-s − 2.37·71-s + 1.28·73-s + 0.227·77-s − 0.337·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3765024446\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3765024446\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.12439743185892429556539573398, −9.860708935230116796945196939023, −9.637483483162028476701941310990, −8.826926585786311223434476879180, −8.339207801866229059117350340323, −8.052555698255183327580179867984, −7.75588840069431215917490103227, −7.25708377973657127048374765194, −7.01396122239580486441216968313, −6.18445510341878905351390689953, −6.11669449839871372112995610996, −5.08873450902929005924555282010, −5.04526338719711150937495835419, −4.67771826047983588679405632832, −3.70422100998641008566470191326, −3.35769667524568380982176608366, −3.14459992634473508449514154849, −2.14243528799439217826877084976, −1.65325479673313526745482323510, −0.27063631687424285128476980581,
0.27063631687424285128476980581, 1.65325479673313526745482323510, 2.14243528799439217826877084976, 3.14459992634473508449514154849, 3.35769667524568380982176608366, 3.70422100998641008566470191326, 4.67771826047983588679405632832, 5.04526338719711150937495835419, 5.08873450902929005924555282010, 6.11669449839871372112995610996, 6.18445510341878905351390689953, 7.01396122239580486441216968313, 7.25708377973657127048374765194, 7.75588840069431215917490103227, 8.052555698255183327580179867984, 8.339207801866229059117350340323, 8.826926585786311223434476879180, 9.637483483162028476701941310990, 9.860708935230116796945196939023, 10.12439743185892429556539573398