| L(s) = 1 | − 4·9-s − 4·17-s − 2·25-s + 2·49-s + 4·73-s + 10·81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4·9-s − 4·17-s − 2·25-s + 2·49-s + 4·73-s + 10·81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1718022968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1718022968\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−7.02422408917260611142478651111, −6.95133679846000348904702588559, −6.64934749368689905267604189069, −6.37512050132226794038680372742, −6.21420698267797631279798655882, −6.10944500377023084003472926039, −5.96936623512845364835318545403, −5.65278155660643339239987732125, −5.37173359460988602407981346933, −5.21404517380726172785640661391, −4.97994506259223041681830704473, −4.77758906319748045592440684855, −4.50102483581328217115477055380, −4.05939879201996381022235652347, −3.86138599817062274226591239329, −3.80143317610510809605380709422, −3.36803280152745760360360494033, −3.15437356169946264646723041268, −2.68253133036454636010506112612, −2.54668252330011589842346845765, −2.18897370204960924819179293151, −2.14279730663332803716379781987, −2.08601991965922016682182788152, −1.06166890326627544015399119234, −0.29217866896677768525668913622,
0.29217866896677768525668913622, 1.06166890326627544015399119234, 2.08601991965922016682182788152, 2.14279730663332803716379781987, 2.18897370204960924819179293151, 2.54668252330011589842346845765, 2.68253133036454636010506112612, 3.15437356169946264646723041268, 3.36803280152745760360360494033, 3.80143317610510809605380709422, 3.86138599817062274226591239329, 4.05939879201996381022235652347, 4.50102483581328217115477055380, 4.77758906319748045592440684855, 4.97994506259223041681830704473, 5.21404517380726172785640661391, 5.37173359460988602407981346933, 5.65278155660643339239987732125, 5.96936623512845364835318545403, 6.10944500377023084003472926039, 6.21420698267797631279798655882, 6.37512050132226794038680372742, 6.64934749368689905267604189069, 6.95133679846000348904702588559, 7.02422408917260611142478651111
