| L(s) = 1 | − 5·9-s + 4·11-s − 16·19-s + 12·29-s − 2·31-s + 12·41-s − 4·49-s + 18·59-s + 20·61-s + 6·71-s − 28·79-s + 9·81-s − 6·89-s − 20·99-s + 24·101-s + 40·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + ⋯ |
| L(s) = 1 | − 5/3·9-s + 1.20·11-s − 3.67·19-s + 2.22·29-s − 0.359·31-s + 1.87·41-s − 4/7·49-s + 2.34·59-s + 2.56·61-s + 0.712·71-s − 3.15·79-s + 81-s − 0.635·89-s − 2.01·99-s + 2.38·101-s + 3.83·109-s + 0.909·121-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 − 4·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.568694726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.568694726\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 846 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 85 T^{2} + 2856 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 6006 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 9 T + 130 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 181 T^{2} + 15312 T^{4} + 181 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 337 T^{2} + 47136 T^{4} + 337 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.15785418336555533605244835119, −5.63596220592554121554653069722, −5.48735977370804065551255828749, −5.34716846700326993552583506077, −5.29463528940873656816800228905, −4.76294002453403777017830063275, −4.75393808083739518255611079249, −4.46263131132669701585907536196, −4.32332960256848921493586343134, −3.98699328410602730381970483555, −3.93613277461613877306480102414, −3.85738657331385750700049087305, −3.63852695119969555239119150797, −3.00034785218012165053470693333, −2.94842817339672607497200936411, −2.78541859379829944572219183271, −2.78407341009622079396080676794, −2.19553695945643343039731327788, −2.08004079805798229675021501766, −1.91617997418576259561054213057, −1.76165330368244171048515843815, −1.17715645263611409629467705286, −0.75665884709567836923902940870, −0.64111603898426997818886381502, −0.34211179336459940434054612351,
0.34211179336459940434054612351, 0.64111603898426997818886381502, 0.75665884709567836923902940870, 1.17715645263611409629467705286, 1.76165330368244171048515843815, 1.91617997418576259561054213057, 2.08004079805798229675021501766, 2.19553695945643343039731327788, 2.78407341009622079396080676794, 2.78541859379829944572219183271, 2.94842817339672607497200936411, 3.00034785218012165053470693333, 3.63852695119969555239119150797, 3.85738657331385750700049087305, 3.93613277461613877306480102414, 3.98699328410602730381970483555, 4.32332960256848921493586343134, 4.46263131132669701585907536196, 4.75393808083739518255611079249, 4.76294002453403777017830063275, 5.29463528940873656816800228905, 5.34716846700326993552583506077, 5.48735977370804065551255828749, 5.63596220592554121554653069722, 6.15785418336555533605244835119
