| L(s) = 1 | − 2-s − 3-s + 4-s + 5·5-s + 6-s − 8-s + 9-s − 5·10-s − 12-s − 5·15-s + 16-s − 18-s + 4·19-s + 5·20-s + 12·23-s + 24-s + 16·25-s − 27-s + 29-s + 5·30-s − 32-s + 36-s − 4·38-s − 5·40-s + 14·43-s + 5·45-s − 12·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 2.23·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.58·10-s − 0.288·12-s − 1.29·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 1.11·20-s + 2.50·23-s + 0.204·24-s + 16/5·25-s − 0.192·27-s + 0.185·29-s + 0.912·30-s − 0.176·32-s + 1/6·36-s − 0.648·38-s − 0.790·40-s + 2.13·43-s + 0.745·45-s − 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.052414559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.052414559\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−8.816283343783412534896950357891, −8.076355990129525126325352712323, −7.51610263927270122997877950052, −7.07352929134273446964466245170, −6.62875704010288800446436344203, −6.23837270111770575374420034931, −5.81627645648565844215449262016, −5.36278450023199675986708951799, −4.91332053096641767655258905766, −4.51596509692775646673157318100, −3.15264004331636899478286342888, −3.03631158629702575356530292846, −2.18289567645718646150189272247, −1.48838224324467285111619123086, −0.982277124950921736803444983401,
0.982277124950921736803444983401, 1.48838224324467285111619123086, 2.18289567645718646150189272247, 3.03631158629702575356530292846, 3.15264004331636899478286342888, 4.51596509692775646673157318100, 4.91332053096641767655258905766, 5.36278450023199675986708951799, 5.81627645648565844215449262016, 6.23837270111770575374420034931, 6.62875704010288800446436344203, 7.07352929134273446964466245170, 7.51610263927270122997877950052, 8.076355990129525126325352712323, 8.816283343783412534896950357891
