| L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 4·8-s + 2·10-s + 5·11-s − 4·13-s + 8·16-s + 8·17-s − 10·19-s + 2·20-s + 10·22-s + 6·23-s − 8·26-s − 5·29-s + 9·31-s + 8·32-s + 16·34-s − 20·37-s − 20·38-s + 4·40-s + 7·41-s + 2·43-s + 10·44-s + 12·46-s + 2·47-s + 7·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 1.41·8-s + 0.632·10-s + 1.50·11-s − 1.10·13-s + 2·16-s + 1.94·17-s − 2.29·19-s + 0.447·20-s + 2.13·22-s + 1.25·23-s − 1.56·26-s − 0.928·29-s + 1.61·31-s + 1.41·32-s + 2.74·34-s − 3.28·37-s − 3.24·38-s + 0.632·40-s + 1.09·41-s + 0.304·43-s + 1.50·44-s + 1.76·46-s + 0.291·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.465658222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.465658222\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 7 T + 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−11.88657537476453472470380889455, −10.93935445073101914610824133097, −10.54949728709666003795157553614, −10.45515097635752729043078404077, −9.543883624788080384083895375648, −9.512393993195809474810266301505, −8.567252287017957271268606885159, −8.380892804581133615615982842053, −7.44701784852275684292512964663, −7.19786071499130443615413172839, −6.75452567382348551595363908591, −5.91207776707295930885432489899, −5.86997298511438985141290507324, −4.98202575537214199687654671054, −4.67611950493511795609400749324, −4.15047423315279489157591034152, −3.54160068514967504957584078213, −2.97748964713865780379152016304, −1.98367875033956582307624119637, −1.35269131782032640114823389683,
1.35269131782032640114823389683, 1.98367875033956582307624119637, 2.97748964713865780379152016304, 3.54160068514967504957584078213, 4.15047423315279489157591034152, 4.67611950493511795609400749324, 4.98202575537214199687654671054, 5.86997298511438985141290507324, 5.91207776707295930885432489899, 6.75452567382348551595363908591, 7.19786071499130443615413172839, 7.44701784852275684292512964663, 8.380892804581133615615982842053, 8.567252287017957271268606885159, 9.512393993195809474810266301505, 9.543883624788080384083895375648, 10.45515097635752729043078404077, 10.54949728709666003795157553614, 10.93935445073101914610824133097, 11.88657537476453472470380889455
