| L(s) = 1 | − 2·5-s − 12·13-s + 2·17-s + 4·19-s − 4·23-s + 5·25-s + 20·29-s − 8·31-s − 6·37-s − 4·41-s − 8·43-s − 8·47-s − 10·53-s − 12·59-s − 2·61-s + 24·65-s − 12·67-s + 24·71-s − 14·73-s + 8·79-s + 24·83-s − 4·85-s + 2·89-s − 8·95-s − 20·97-s + 6·101-s + 2·109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 3.32·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 25-s + 3.71·29-s − 1.43·31-s − 0.986·37-s − 0.624·41-s − 1.21·43-s − 1.16·47-s − 1.37·53-s − 1.56·59-s − 0.256·61-s + 2.97·65-s − 1.46·67-s + 2.84·71-s − 1.63·73-s + 0.900·79-s + 2.63·83-s − 0.433·85-s + 0.211·89-s − 0.820·95-s − 2.03·97-s + 0.597·101-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2566383547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2566383547\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 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
−8.640999272341286951049326553436, −8.154117198051418666831477395359, −8.039157493234952738651435269704, −7.71084010956113871197564615937, −7.31666385090932706879795966349, −6.86697632945751690128319984656, −6.73219651372610855731367190477, −6.33412977954345192539242047627, −5.59848273141288266590724538708, −5.13972949171431704581934301483, −4.90874190317743168797615358570, −4.65586883738171795913397678717, −4.37328711784558422094157111598, −3.44944934689807826946092778836, −3.26544837931114062552187486371, −2.88148956732844158701413511855, −2.35005770868575698380149320852, −1.81349383031692935641239444720, −1.08633607958420632713005377265, −0.16400404974601194804692622465,
0.16400404974601194804692622465, 1.08633607958420632713005377265, 1.81349383031692935641239444720, 2.35005770868575698380149320852, 2.88148956732844158701413511855, 3.26544837931114062552187486371, 3.44944934689807826946092778836, 4.37328711784558422094157111598, 4.65586883738171795913397678717, 4.90874190317743168797615358570, 5.13972949171431704581934301483, 5.59848273141288266590724538708, 6.33412977954345192539242047627, 6.73219651372610855731367190477, 6.86697632945751690128319984656, 7.31666385090932706879795966349, 7.71084010956113871197564615937, 8.039157493234952738651435269704, 8.154117198051418666831477395359, 8.640999272341286951049326553436
