| L(s) = 1 | + 3-s + 7-s + 19-s + 21-s + 25-s − 27-s + 31-s + 37-s + 57-s − 3·67-s − 3·73-s + 75-s − 3·79-s − 81-s + 93-s + 103-s − 109-s + 111-s + 121-s + 127-s + 131-s + 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 3-s + 7-s + 19-s + 21-s + 25-s − 27-s + 31-s + 37-s + 57-s − 3·67-s − 3·73-s + 75-s − 3·79-s − 81-s + 93-s + 103-s − 109-s + 111-s + 121-s + 127-s + 131-s + 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.742150932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742150932\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.883361188035373906419441087587, −9.605770795477014076635233566986, −9.052404586130416601736110748213, −8.626113634243279066258617216375, −8.609628081706607910401976907680, −8.025229439064058784395647123513, −7.57766102744314241722255790277, −7.38454495686700454911791844883, −6.94661736959813500805900863679, −6.23852075790183689620468265926, −5.78329059058980388254034240801, −5.54313069208453472758614419187, −4.67656229756139373130079230119, −4.59252321772901172097261051145, −4.08615738087823309013718195764, −3.21550602071566993232049331248, −2.98334743462291783556861252951, −2.56209811985790774024556318593, −1.68917018632820173520831839496, −1.24239805936841830342557562929,
1.24239805936841830342557562929, 1.68917018632820173520831839496, 2.56209811985790774024556318593, 2.98334743462291783556861252951, 3.21550602071566993232049331248, 4.08615738087823309013718195764, 4.59252321772901172097261051145, 4.67656229756139373130079230119, 5.54313069208453472758614419187, 5.78329059058980388254034240801, 6.23852075790183689620468265926, 6.94661736959813500805900863679, 7.38454495686700454911791844883, 7.57766102744314241722255790277, 8.025229439064058784395647123513, 8.609628081706607910401976907680, 8.626113634243279066258617216375, 9.052404586130416601736110748213, 9.605770795477014076635233566986, 9.883361188035373906419441087587
