| L(s) = 1 | + 2·4-s + 5·7-s + 13-s + 8·19-s − 25-s + 10·28-s + 2·31-s + 2·37-s + 9·43-s + 7·49-s + 2·52-s − 3·61-s − 8·64-s − 15·67-s − 14·73-s + 16·76-s + 6·79-s + 5·91-s + 7·97-s − 2·100-s − 10·103-s + 10·109-s − 5·121-s + 4·124-s + 127-s + 131-s + 40·133-s + ⋯ |
| L(s) = 1 | + 4-s + 1.88·7-s + 0.277·13-s + 1.83·19-s − 1/5·25-s + 1.88·28-s + 0.359·31-s + 0.328·37-s + 1.37·43-s + 49-s + 0.277·52-s − 0.384·61-s − 64-s − 1.83·67-s − 1.63·73-s + 1.83·76-s + 0.675·79-s + 0.524·91-s + 0.710·97-s − 1/5·100-s − 0.985·103-s + 0.957·109-s − 0.454·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 3.46·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.273778522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.273778522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.764346545926302669413322149241, −8.116800026552764309039415190025, −7.75656422768351241565787721705, −7.34477584132857413160870749951, −7.18951135720813301528886215636, −6.25245532873071258354239464576, −6.00945780119819937833215431025, −5.37536773860380432407568736451, −4.86067628904138086549907576434, −4.49248596832030498227268073748, −3.75921844971558256451514556987, −2.98555704204796575751227839603, −2.46250087092456269936239218160, −1.64967964403550599417243681034, −1.19244256623416350094330421936,
1.19244256623416350094330421936, 1.64967964403550599417243681034, 2.46250087092456269936239218160, 2.98555704204796575751227839603, 3.75921844971558256451514556987, 4.49248596832030498227268073748, 4.86067628904138086549907576434, 5.37536773860380432407568736451, 6.00945780119819937833215431025, 6.25245532873071258354239464576, 7.18951135720813301528886215636, 7.34477584132857413160870749951, 7.75656422768351241565787721705, 8.116800026552764309039415190025, 8.764346545926302669413322149241
