| L(s) = 1 | + 3·2-s + 4·4-s + 3·8-s − 9-s + 6·11-s + 3·16-s + 6·17-s − 3·18-s + 6·19-s + 18·22-s − 12·23-s − 5·25-s + 10·31-s + 6·32-s + 18·34-s − 4·36-s + 4·37-s + 18·38-s − 16·43-s + 24·44-s − 36·46-s − 6·47-s − 15·50-s − 6·53-s + 6·59-s − 6·61-s + 30·62-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s + 1.06·8-s − 1/3·9-s + 1.80·11-s + 3/4·16-s + 1.45·17-s − 0.707·18-s + 1.37·19-s + 3.83·22-s − 2.50·23-s − 25-s + 1.79·31-s + 1.06·32-s + 3.08·34-s − 2/3·36-s + 0.657·37-s + 2.91·38-s − 2.43·43-s + 3.61·44-s − 5.30·46-s − 0.875·47-s − 2.12·50-s − 0.824·53-s + 0.781·59-s − 0.768·61-s + 3.81·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.37816218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.37816218\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 173 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.982820559881118595989684120727, −7.73496349705862751558404028915, −7.03618598381248319923111028360, −6.69005494630249353699739735354, −6.41351457488829962799862626931, −6.20328545158585072117424451383, −5.65978229613637070655340511471, −5.62405833519709966136408854330, −5.12242155758843402766723540319, −4.89862679230168212830244837030, −4.21939438976793778270733002014, −4.17052913989804499796135433640, −3.83523993381332403445586364991, −3.44986878532506369747052712632, −3.00538890566313823792091989657, −2.94719526213056545435382393792, −2.01256227550936978261495537370, −1.66539717911064327220235980572, −1.24658112869423572569226566663, −0.54379000206957633115181158789,
0.54379000206957633115181158789, 1.24658112869423572569226566663, 1.66539717911064327220235980572, 2.01256227550936978261495537370, 2.94719526213056545435382393792, 3.00538890566313823792091989657, 3.44986878532506369747052712632, 3.83523993381332403445586364991, 4.17052913989804499796135433640, 4.21939438976793778270733002014, 4.89862679230168212830244837030, 5.12242155758843402766723540319, 5.62405833519709966136408854330, 5.65978229613637070655340511471, 6.20328545158585072117424451383, 6.41351457488829962799862626931, 6.69005494630249353699739735354, 7.03618598381248319923111028360, 7.73496349705862751558404028915, 7.982820559881118595989684120727
