| L(s) = 1 | − 243·3-s − 4.75e3·7-s + 3.93e4·9-s + 9.62e5·19-s + 1.15e6·21-s + 1.95e6·25-s − 4.78e6·27-s − 1.13e7·31-s + 2.22e7·37-s − 8.87e7·43-s − 1.77e7·49-s − 2.33e8·57-s − 3.14e8·61-s − 1.87e8·63-s − 2.12e8·67-s + 1.11e8·73-s − 4.74e8·75-s − 5.80e8·79-s + 3.87e8·81-s + 2.75e9·93-s + 2.07e9·103-s + 5.38e8·109-s − 5.41e9·111-s − 2.35e9·121-s + 127-s + 2.15e10·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.747·7-s + 2·9-s + 1.69·19-s + 1.29·21-s + 25-s − 1.73·27-s − 2.20·31-s + 1.95·37-s − 3.95·43-s − 0.440·49-s − 2.93·57-s − 2.90·61-s − 1.49·63-s − 1.28·67-s + 0.460·73-s − 1.73·75-s − 1.67·79-s + 81-s + 3.81·93-s + 1.81·103-s + 0.365·109-s − 3.38·111-s − 121-s + 6.85·129-s − 1.26·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2385736023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2385736023\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2$ | \( 1 + p^{5} T + p^{9} T^{2} \) |
| 7 | $C_2$ | \( 1 + 4751 T + p^{9} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{9} T^{2} + p^{18} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 86777 T + p^{9} T^{2} )( 1 + 86777 T + p^{9} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 976696 T + p^{9} T^{2} )( 1 + 14257 T + p^{9} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p^{9} T^{2} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 1691228 T + p^{9} T^{2} )( 1 + 9630469 T + p^{9} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 15384490 T + p^{9} T^{2} )( 1 - 6880789 T + p^{9} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 44367317 T + p^{9} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{9} T^{2} + p^{18} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 98072641 T + p^{9} T^{2} )( 1 + 215975699 T + p^{9} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 112542320 T + p^{9} T^{2} )( 1 + 324823339 T + p^{9} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 296368310 T + p^{9} T^{2} )( 1 + 184590737 T + p^{9} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 35812097 T + p^{9} T^{2} )( 1 + 616732324 T + p^{9} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{9} T^{2} + p^{18} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1288928270 T + p^{9} T^{2} )( 1 + 1288928270 T + p^{9} T^{2} ) \) |
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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
−12.74611572061844724765878414109, −11.85192657541982617635289025391, −11.79388630567529041339800439660, −11.08014406908069257263369054471, −10.67896152464297296073311773591, −9.944187768489869820179234293275, −9.641109927040952710663101504783, −9.000138513261794635703228697229, −8.060886620180625663957522922308, −7.13081745527822132451594714481, −7.09116460545867488895286380037, −6.00644908436319956722028893586, −5.94830983623849222014842912984, −4.91448845450942213766067749873, −4.75071750251797066537001643750, −3.52530305989371014493735343989, −3.06539109747122428268665363175, −1.67497427874651652427246228423, −1.12131124035316696885194427809, −0.17438949821868922097780064242,
0.17438949821868922097780064242, 1.12131124035316696885194427809, 1.67497427874651652427246228423, 3.06539109747122428268665363175, 3.52530305989371014493735343989, 4.75071750251797066537001643750, 4.91448845450942213766067749873, 5.94830983623849222014842912984, 6.00644908436319956722028893586, 7.09116460545867488895286380037, 7.13081745527822132451594714481, 8.060886620180625663957522922308, 9.000138513261794635703228697229, 9.641109927040952710663101504783, 9.944187768489869820179234293275, 10.67896152464297296073311773591, 11.08014406908069257263369054471, 11.79388630567529041339800439660, 11.85192657541982617635289025391, 12.74611572061844724765878414109
