| L(s) = 1 | − 3-s + 2·5-s − 2·9-s + 2·11-s + 8·13-s − 2·15-s − 4·16-s − 4·17-s − 23-s − 7·25-s + 5·27-s + 14·31-s − 2·33-s − 8·39-s − 4·45-s + 4·48-s − 10·49-s + 4·51-s − 12·53-s + 4·55-s + 16·65-s + 69-s + 8·73-s + 7·75-s − 8·80-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 2/3·9-s + 0.603·11-s + 2.21·13-s − 0.516·15-s − 16-s − 0.970·17-s − 0.208·23-s − 7/5·25-s + 0.962·27-s + 2.51·31-s − 0.348·33-s − 1.28·39-s − 0.596·45-s + 0.577·48-s − 1.42·49-s + 0.560·51-s − 1.64·53-s + 0.539·55-s + 1.98·65-s + 0.120·69-s + 0.936·73-s + 0.808·75-s − 0.894·80-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.745357748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.745357748\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.603539619290756001226038948684, −8.066676450133041185875497615415, −7.63007897940884090788433437207, −6.71063709122250323132785439185, −6.36261389471308870138602900888, −6.21596253416335450587403773340, −5.98564341355583648065538713444, −5.22817631161736866028777591734, −4.63121926718722083713568972634, −4.30607886187937345087959060428, −3.53762232830670711802398982336, −3.06181471886256813930137795208, −2.15954006937838202982985808129, −1.70386697519836408216378877863, −0.71341471544986268205062806263,
0.71341471544986268205062806263, 1.70386697519836408216378877863, 2.15954006937838202982985808129, 3.06181471886256813930137795208, 3.53762232830670711802398982336, 4.30607886187937345087959060428, 4.63121926718722083713568972634, 5.22817631161736866028777591734, 5.98564341355583648065538713444, 6.21596253416335450587403773340, 6.36261389471308870138602900888, 6.71063709122250323132785439185, 7.63007897940884090788433437207, 8.066676450133041185875497615415, 8.603539619290756001226038948684
