| L(s) = 1 | + 2.94e3·4-s + 9.37e5·11-s + 4.44e6·16-s + 7.58e5·19-s + 1.39e8·29-s + 3.42e8·31-s − 3.82e8·41-s + 2.75e9·44-s + 3.64e9·49-s + 1.85e9·59-s − 2.17e10·61-s + 7.49e8·64-s + 4.59e10·71-s + 2.23e9·76-s + 4.15e10·79-s + 1.26e11·89-s − 6.57e10·101-s + 3.32e11·109-s + 4.09e11·116-s + 8.86e10·121-s + 1.00e12·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.43·4-s + 1.75·11-s + 1.06·16-s + 0.0703·19-s + 1.26·29-s + 2.15·31-s − 0.515·41-s + 2.51·44-s + 1.84·49-s + 0.337·59-s − 3.30·61-s + 0.0872·64-s + 3.02·71-s + 0.100·76-s + 1.51·79-s + 2.39·89-s − 0.622·101-s + 2.06·109-s + 1.81·116-s + 0.310·121-s + 3.08·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(9.147660967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.147660967\) |
| \(L(\frac{13}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 735 p^{2} T^{2} + p^{22} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 74417150 p^{2} T^{2} + p^{22} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 468788 T + p^{11} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3444413370310 T^{2} + p^{22} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 54673486405470 T^{2} + p^{22} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 379460 T + p^{11} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 852091779547390 T^{2} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 69696710 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 171448632 T + p^{11} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 270955929815342710 T^{2} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 191343242 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 1238510561234596250 T^{2} + p^{22} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2308663836035458830 T^{2} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 18122190498163779030 T^{2} + p^{22} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 925569220 T + p^{11} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10898589338 T + p^{11} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - \)\(22\!\cdots\!70\)\( T^{2} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 22966943728 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - \)\(52\!\cdots\!90\)\( T^{2} + p^{22} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 20768886240 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - \)\(25\!\cdots\!70\)\( T^{2} + p^{22} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 63176321130 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + \)\(16\!\cdots\!70\)\( T^{2} + p^{22} T^{4} \) |
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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
−10.61921560795186056267024799979, −10.20263938595506304731164461807, −9.441148596522100577159613725891, −9.228688893309457541910115718214, −8.494516405132351907163737405973, −8.055405028171616443499515358311, −7.52076933974139008109374287875, −6.82669395112977679082265700392, −6.63539860469919004491352015119, −6.21601378029021544673146000707, −5.79250806607480924031622821976, −4.78614980631129570789448422622, −4.52907083307544702120402469411, −3.65274118774266395283472719649, −3.31598488453778144612105497702, −2.54303800424086385370996950014, −2.21846453531789569874999858891, −1.48729636064225494344051705010, −1.02036177396865865083428663933, −0.61165366392120280322022220269,
0.61165366392120280322022220269, 1.02036177396865865083428663933, 1.48729636064225494344051705010, 2.21846453531789569874999858891, 2.54303800424086385370996950014, 3.31598488453778144612105497702, 3.65274118774266395283472719649, 4.52907083307544702120402469411, 4.78614980631129570789448422622, 5.79250806607480924031622821976, 6.21601378029021544673146000707, 6.63539860469919004491352015119, 6.82669395112977679082265700392, 7.52076933974139008109374287875, 8.055405028171616443499515358311, 8.494516405132351907163737405973, 9.228688893309457541910115718214, 9.441148596522100577159613725891, 10.20263938595506304731164461807, 10.61921560795186056267024799979
