| L(s) = 1 | − 2-s − 4-s − 7-s + 8-s − 2·11-s − 3·13-s + 14-s − 16-s + 17-s + 6·19-s + 2·22-s + 2·23-s + 6·25-s + 3·26-s + 28-s − 9·29-s + 6·31-s + 5·32-s − 34-s − 6·38-s − 16·41-s + 5·43-s + 2·44-s − 2·46-s + 5·47-s − 4·49-s − 6·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s − 0.832·13-s + 0.267·14-s − 1/4·16-s + 0.242·17-s + 1.37·19-s + 0.426·22-s + 0.417·23-s + 6/5·25-s + 0.588·26-s + 0.188·28-s − 1.67·29-s + 1.07·31-s + 0.883·32-s − 0.171·34-s − 0.973·38-s − 2.49·41-s + 0.762·43-s + 0.301·44-s − 0.294·46-s + 0.729·47-s − 4/7·49-s − 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 763 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 763 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3048575004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3048575004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 109 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 10 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 5 T - 10 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 172 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 88 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−18.9583811480, −18.7483825331, −18.4160275264, −17.6518024534, −17.0473635098, −16.7648007594, −15.7640521811, −15.4834175137, −14.5456362037, −14.0360259295, −13.2107438097, −12.8352645583, −11.9057314678, −11.3330526293, −10.2367664765, −9.87244560332, −9.14464265480, −8.50691630928, −7.58102252001, −6.89245554213, −5.55537958997, −4.71372410195, −3.09273629803,
3.09273629803, 4.71372410195, 5.55537958997, 6.89245554213, 7.58102252001, 8.50691630928, 9.14464265480, 9.87244560332, 10.2367664765, 11.3330526293, 11.9057314678, 12.8352645583, 13.2107438097, 14.0360259295, 14.5456362037, 15.4834175137, 15.7640521811, 16.7648007594, 17.0473635098, 17.6518024534, 18.4160275264, 18.7483825331, 18.9583811480
