| L(s) = 1 | + 4·7-s − 16-s + 8·49-s − 4·73-s + 4·97-s − 4·103-s − 4·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 4·7-s − 16-s + 8·49-s − 4·73-s + 4·97-s − 4·103-s − 4·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.217956324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.217956324\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
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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
−6.69106783609463915421471842486, −6.60424128555511810470876836844, −6.58048612104266458918414747684, −6.11542491827467045843340964952, −5.83666076838975543748442152919, −5.61301001469599497995527214091, −5.47740960252324958243663073236, −5.19778009045072591400710342222, −5.10219579332562101002688585285, −4.97824090418571031361734352697, −4.49913603736955025405778657181, −4.36871636935297624993249702586, −4.33701070402582716012451235259, −4.18481187228071016322039162884, −3.98755389708202670470417382156, −3.28837668714511074863237829324, −3.23167243811718331006799878602, −2.97085972300348218395614389696, −2.45130827950648851578163056609, −2.36646873617890536542613806966, −1.97089657785380041480085427782, −1.69108136230916245936672200532, −1.64056384781367897080050106323, −1.25857026903807500460493415343, −0.849184209890840376901927111587,
0.849184209890840376901927111587, 1.25857026903807500460493415343, 1.64056384781367897080050106323, 1.69108136230916245936672200532, 1.97089657785380041480085427782, 2.36646873617890536542613806966, 2.45130827950648851578163056609, 2.97085972300348218395614389696, 3.23167243811718331006799878602, 3.28837668714511074863237829324, 3.98755389708202670470417382156, 4.18481187228071016322039162884, 4.33701070402582716012451235259, 4.36871636935297624993249702586, 4.49913603736955025405778657181, 4.97824090418571031361734352697, 5.10219579332562101002688585285, 5.19778009045072591400710342222, 5.47740960252324958243663073236, 5.61301001469599497995527214091, 5.83666076838975543748442152919, 6.11542491827467045843340964952, 6.58048612104266458918414747684, 6.60424128555511810470876836844, 6.69106783609463915421471842486
