| L(s) = 1 | + 2·7-s − 6·19-s + 4·25-s − 6·31-s + 2·37-s + 4·43-s − 11·49-s − 12·61-s + 22·67-s + 6·73-s + 10·79-s + 30·103-s + 2·109-s − 20·121-s + 127-s + 131-s − 12·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 8·175-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.37·19-s + 4/5·25-s − 1.07·31-s + 0.328·37-s + 0.609·43-s − 1.57·49-s − 1.53·61-s + 2.68·67-s + 0.702·73-s + 1.12·79-s + 2.95·103-s + 0.191·109-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s − 1.04·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + 0.604·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.495721225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.495721225\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 20 T^{2} + 279 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 56 T^{2} + 927 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 98 T^{2} + 6795 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 94 T^{2} + 5355 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 154 T^{2} + 15795 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.24343485790854955537990323303, −6.78995797602136760161903947639, −6.53264035431712112126521028713, −6.53210884034060517273424955987, −6.32976192454681345308502559923, −6.18265152210481856790290321320, −5.62625932059164759360601788020, −5.61836615368041125296047072665, −5.14849547405912801082018804465, −5.03176727449148042668527168400, −4.98075336179054579389566988502, −4.66543206211386908994888247897, −4.21669067850493467362631780918, −4.08485114690419334502937453661, −3.93858783417101216152425231948, −3.55260931380509967147128746600, −3.31402426335908474762141077242, −2.95448379391483643644199071445, −2.63135504273872976414255921917, −2.40036120951079668965161460458, −1.97408396542527041998296691842, −1.75572287177399794240998431593, −1.46807278329741051670278782812, −0.861898717497894710039953335092, −0.40867639096624215351846639502,
0.40867639096624215351846639502, 0.861898717497894710039953335092, 1.46807278329741051670278782812, 1.75572287177399794240998431593, 1.97408396542527041998296691842, 2.40036120951079668965161460458, 2.63135504273872976414255921917, 2.95448379391483643644199071445, 3.31402426335908474762141077242, 3.55260931380509967147128746600, 3.93858783417101216152425231948, 4.08485114690419334502937453661, 4.21669067850493467362631780918, 4.66543206211386908994888247897, 4.98075336179054579389566988502, 5.03176727449148042668527168400, 5.14849547405912801082018804465, 5.61836615368041125296047072665, 5.62625932059164759360601788020, 6.18265152210481856790290321320, 6.32976192454681345308502559923, 6.53210884034060517273424955987, 6.53264035431712112126521028713, 6.78995797602136760161903947639, 7.24343485790854955537990323303
