L(s) = 1 | + 5·5-s + 10·25-s + 5·49-s − 5·59-s + 5·67-s − 5·103-s + 10·125-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 + ⋯ |
L(s) = 1 | + 5·5-s + 10·25-s + 5·49-s − 5·59-s + 5·67-s − 5·103-s + 10·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{60} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{60} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.663118891\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.663118891\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5} \) |
| 11 | \( 1 - T^{5} + T^{10} - T^{15} + T^{20} \) |
good | 2 | \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \) |
| 3 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 7 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5} \) |
| 13 | \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \) |
| 17 | \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \) |
| 19 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 23 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 29 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 31 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \) |
| 37 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 41 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 43 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5} \) |
| 47 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 53 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 59 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 61 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 67 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 71 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 73 | \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \) |
| 79 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 83 | \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \) |
| 89 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \) |
| 97 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.41287710661874078143059109565, −2.29732581777403436243202925782, −2.24707115476144043199740900202, −2.22874527982273044802854840468, −2.15697220720568276998716105515, −2.15289711649007428283095471935, −1.99275620702869840469499402642, −1.86313076305075860826946815005, −1.83028878945929911818838507639, −1.79505804774312712030162403029, −1.68999365872134201553695082455, −1.60944277667116818686008988413, −1.51951211433278030176650159843, −1.45775368225439779630557728360, −1.43857664123639013649010175916, −1.43138466614939977835735629254, −1.40422919559022059353558344525, −1.35781071947619600870961983824, −1.13214874854336410026036614170, −1.08230890728316459557230742704, −1.01026032627064722823017040234, −0.998537371765420855711265507666, −0.76528468281201838721013149177, −0.65010933790305929851375691188, −0.49724346505651285665227898390,
0.49724346505651285665227898390, 0.65010933790305929851375691188, 0.76528468281201838721013149177, 0.998537371765420855711265507666, 1.01026032627064722823017040234, 1.08230890728316459557230742704, 1.13214874854336410026036614170, 1.35781071947619600870961983824, 1.40422919559022059353558344525, 1.43138466614939977835735629254, 1.43857664123639013649010175916, 1.45775368225439779630557728360, 1.51951211433278030176650159843, 1.60944277667116818686008988413, 1.68999365872134201553695082455, 1.79505804774312712030162403029, 1.83028878945929911818838507639, 1.86313076305075860826946815005, 1.99275620702869840469499402642, 2.15289711649007428283095471935, 2.15697220720568276998716105515, 2.22874527982273044802854840468, 2.24707115476144043199740900202, 2.29732581777403436243202925782, 2.41287710661874078143059109565
Plot not available for L-functions of degree greater than 10.