| L(s) = 1 | − 2.04e3·4-s − 5.90e4·9-s + 3.14e6·16-s + 4.04e6·19-s − 9.96e7·31-s + 1.20e8·36-s − 5.23e8·49-s + 3.10e9·61-s − 4.29e9·64-s − 8.29e9·76-s − 7.91e9·79-s + 3.48e9·81-s + 2.59e10·109-s + 5.18e10·121-s + 2.04e11·124-s + 127-s + 131-s + 137-s + 139-s − 1.85e11·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.55e11·169-s − 2.39e11·171-s + ⋯ |
| L(s) = 1 | − 2·4-s − 9-s + 3·16-s + 1.63·19-s − 3.48·31-s + 2·36-s − 1.85·49-s + 3.67·61-s − 4·64-s − 3.27·76-s − 2.57·79-s + 81-s + 1.68·109-s + 2·121-s + 6.96·124-s − 3·144-s + 1.85·169-s − 1.63·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.5527251803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5527251803\) |
| \(L(6)\) |
|
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 | $C_2$ | \( 1 + p^{10} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 523323623 T^{2} + p^{20} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 255564058177 T^{2} + p^{20} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2024677 T + p^{10} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 49843573 T + p^{10} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8665815522315698 T^{2} + p^{20} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 1343935055601623 T^{2} + p^{20} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 1551490727 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 3635284414544796023 T^{2} + p^{20} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 8577821547816235298 T^{2} + p^{20} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3959005298 T + p^{10} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 60206365955391250223 T^{2} + p^{20} T^{4} \) |
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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
−13.15158434880079946682526515731, −12.44227828021837490822790332131, −11.43816525585159936018294299980, −11.31847839512327052899950184202, −10.31840093563244082040483838402, −9.701917104157232046780090420049, −9.418444708666897321080609189665, −8.729165330718647434243663391953, −8.456153970065773958690008961929, −7.67173831820330857093176255201, −7.14803118640107800145416333052, −5.93025046228471738434746354154, −5.29924500367942573690073964135, −5.24771893066035342248578211490, −4.21957887390812707165605198897, −3.51823187249988215508611496191, −3.17428161738597523367156649791, −1.86630994564586517446452313065, −0.953091616652870910559610826400, −0.26290615618916544829109700957,
0.26290615618916544829109700957, 0.953091616652870910559610826400, 1.86630994564586517446452313065, 3.17428161738597523367156649791, 3.51823187249988215508611496191, 4.21957887390812707165605198897, 5.24771893066035342248578211490, 5.29924500367942573690073964135, 5.93025046228471738434746354154, 7.14803118640107800145416333052, 7.67173831820330857093176255201, 8.456153970065773958690008961929, 8.729165330718647434243663391953, 9.418444708666897321080609189665, 9.701917104157232046780090420049, 10.31840093563244082040483838402, 11.31847839512327052899950184202, 11.43816525585159936018294299980, 12.44227828021837490822790332131, 13.15158434880079946682526515731
