L(s) = 1 | − 44·31-s − 4·61-s − 9·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
L(s) = 1 | − 7.90·31-s − 0.512·61-s − 81-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9688981079\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9688981079\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^3$ | \( 1 + 71 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 191 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 2062 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 3191 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 8809 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 8542 T^{4} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 9071 T^{4} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.822555355197016147019460866317, −8.166704774242188692017644598861, −7.953120423714196314595464752976, −7.84148658730117904538530503067, −7.43842888431618224300548340974, −7.27063806508378292236530837756, −7.00924955764891364215982017054, −6.88498259536766895505327696159, −6.62190355994087854172886235568, −6.06431871183911066777028233098, −5.71177491798444827396276226959, −5.52482853605190452122363885586, −5.45229441999026992233723255918, −5.38283383068591988890539478739, −4.60886312772752862324915921247, −4.44020035648509998984483148507, −4.15313930557551341846262812536, −3.61653648692917148728047209142, −3.49490087929753217566214985753, −3.29679380227759339369699130625, −2.80015991950373825266521744282, −2.00105029732420890941231738876, −1.78628838782571669735172043363, −1.77973688557852886147883827335, −0.43487692495510207052947178929,
0.43487692495510207052947178929, 1.77973688557852886147883827335, 1.78628838782571669735172043363, 2.00105029732420890941231738876, 2.80015991950373825266521744282, 3.29679380227759339369699130625, 3.49490087929753217566214985753, 3.61653648692917148728047209142, 4.15313930557551341846262812536, 4.44020035648509998984483148507, 4.60886312772752862324915921247, 5.38283383068591988890539478739, 5.45229441999026992233723255918, 5.52482853605190452122363885586, 5.71177491798444827396276226959, 6.06431871183911066777028233098, 6.62190355994087854172886235568, 6.88498259536766895505327696159, 7.00924955764891364215982017054, 7.27063806508378292236530837756, 7.43842888431618224300548340974, 7.84148658730117904538530503067, 7.953120423714196314595464752976, 8.166704774242188692017644598861, 8.822555355197016147019460866317