| L(s) = 1 | + 4·9-s + 2·11-s + 8·23-s + 10·25-s + 8·29-s + 8·37-s + 8·43-s + 8·53-s + 16·67-s + 20·79-s + 9·81-s + 8·99-s + 24·107-s + 20·109-s + 56·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 0.603·11-s + 1.66·23-s + 2·25-s + 1.48·29-s + 1.31·37-s + 1.21·43-s + 1.09·53-s + 1.95·67-s + 2.25·79-s + 81-s + 0.804·99-s + 2.32·107-s + 1.91·109-s + 5.26·113-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.31134814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.31134814\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 16 T^{2} - 33 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 44 T^{2} + 975 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 4 T^{2} - 2193 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 100 T^{2} + 6519 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 40 T^{2} - 2121 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 144 T^{2} + 15407 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 50 T^{2} - 5421 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.61029443398192901305017912042, −6.19913271860185442274621953720, −6.12499582334188336426202107904, −5.83802496510041860472718140662, −5.73895242078462659290465345690, −5.18517289328380608426318316847, −5.10263312365293997878856193351, −5.00481534702537304736312482120, −4.82547367399436220357001523186, −4.43854773878332127387637844526, −4.24742316397698469615886274129, −4.16312006777725377835686032411, −4.13894262629125312046146407048, −3.31553234841752865066648180006, −3.24999023558210615085528883730, −3.24532167763463205435077415999, −3.18501656123145836384266699957, −2.41499464965090300152443686101, −2.34704173477180368403423305598, −2.07962606377655548064038089994, −1.91967395231479030561324067062, −1.20921582178916836626915820982, −0.984167329389546213044403349925, −0.810889429789893705305067641421, −0.72919177675003228043815857496,
0.72919177675003228043815857496, 0.810889429789893705305067641421, 0.984167329389546213044403349925, 1.20921582178916836626915820982, 1.91967395231479030561324067062, 2.07962606377655548064038089994, 2.34704173477180368403423305598, 2.41499464965090300152443686101, 3.18501656123145836384266699957, 3.24532167763463205435077415999, 3.24999023558210615085528883730, 3.31553234841752865066648180006, 4.13894262629125312046146407048, 4.16312006777725377835686032411, 4.24742316397698469615886274129, 4.43854773878332127387637844526, 4.82547367399436220357001523186, 5.00481534702537304736312482120, 5.10263312365293997878856193351, 5.18517289328380608426318316847, 5.73895242078462659290465345690, 5.83802496510041860472718140662, 6.12499582334188336426202107904, 6.19913271860185442274621953720, 6.61029443398192901305017912042
