L(s) = 1 | + 2·5-s + 2·13-s + 4·17-s + 2·25-s + 6·29-s − 10·37-s + 14·49-s + 18·53-s − 2·61-s + 4·65-s + 8·85-s − 16·97-s + 18·101-s + 14·109-s + 32·113-s + 10·125-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.554·13-s + 0.970·17-s + 2/5·25-s + 1.11·29-s − 1.64·37-s + 2·49-s + 2.47·53-s − 0.256·61-s + 0.496·65-s + 0.867·85-s − 1.62·97-s + 1.79·101-s + 1.34·109-s + 3.01·113-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.996·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.450461524\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.450461524\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.495990798371836682178950568105, −8.350729149007756589859324713126, −7.66481479287116849880675349808, −7.42131480778202259531747235694, −6.91438854938860765028197656955, −6.86882153217375179451994709921, −6.18132580012906941399980901513, −5.92745409209069382125638039820, −5.59013336230329036198595987274, −5.35194630713359224657037072365, −4.79940927341321286917146669003, −4.48973493160142592018787505122, −3.79806402014332806114181623906, −3.69588338593725878564570192262, −2.98295216442246247129436463320, −2.77162956573359561061413093783, −1.98154852064004869687644201864, −1.86768030491735816623022548208, −0.995432605139636778114986141510, −0.69348739381659912831829692192,
0.69348739381659912831829692192, 0.995432605139636778114986141510, 1.86768030491735816623022548208, 1.98154852064004869687644201864, 2.77162956573359561061413093783, 2.98295216442246247129436463320, 3.69588338593725878564570192262, 3.79806402014332806114181623906, 4.48973493160142592018787505122, 4.79940927341321286917146669003, 5.35194630713359224657037072365, 5.59013336230329036198595987274, 5.92745409209069382125638039820, 6.18132580012906941399980901513, 6.86882153217375179451994709921, 6.91438854938860765028197656955, 7.42131480778202259531747235694, 7.66481479287116849880675349808, 8.350729149007756589859324713126, 8.495990798371836682178950568105