| L(s) = 1 | + 5·9-s + 4·11-s − 4·19-s + 18·29-s − 12·31-s − 8·41-s + 9·49-s − 28·59-s − 10·61-s − 4·71-s − 22·79-s + 4·81-s − 14·89-s + 20·99-s + 34·101-s − 10·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 1.20·11-s − 0.917·19-s + 3.34·29-s − 2.15·31-s − 1.24·41-s + 9/7·49-s − 3.64·59-s − 1.28·61-s − 0.474·71-s − 2.47·79-s + 4/9·81-s − 1.48·89-s + 2.01·99-s + 3.38·101-s − 0.957·109-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 + 2/13·169-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
| \(L(1)\) |
\(\approx\) |
\(1.273449189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273449189\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 7 p T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 9 T^{2} + 37 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 321 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 821 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 9 T + 49 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1491 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 205 T^{2} + 16121 T^{4} - 205 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 221 T^{2} + 22137 T^{4} - 221 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 11 T + 159 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 109 T^{2} + 6917 T^{4} - 109 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 7 T + 187 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 17 T^{2} + 16521 T^{4} - 17 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.86180425310284841242924383360, −5.79541986940834833416421878507, −5.75929933302243275591941041872, −5.13453185508180892564053720610, −4.95636910060113215603955484263, −4.87475472329821282395741608023, −4.79026280665010030185415953282, −4.39015237267620016071973132426, −4.29948723498312921359661198126, −4.19207400774589290494702632423, −4.11717817004201581867908733121, −3.63509263252700718754121583203, −3.37497581786662752671429438514, −3.37308117511859779474210871641, −3.14394703366650379974078012920, −2.79736317887975206199752263782, −2.52173082535976043270798528711, −2.34768618212127408704412799144, −2.03464063972095173927573005295, −1.54911034786201867711883050409, −1.52648357535240705285244676822, −1.44422867953277196557160256412, −1.10382496678185013011507823931, −0.68386610627091710063430171487, −0.14525844383097415859734621194,
0.14525844383097415859734621194, 0.68386610627091710063430171487, 1.10382496678185013011507823931, 1.44422867953277196557160256412, 1.52648357535240705285244676822, 1.54911034786201867711883050409, 2.03464063972095173927573005295, 2.34768618212127408704412799144, 2.52173082535976043270798528711, 2.79736317887975206199752263782, 3.14394703366650379974078012920, 3.37308117511859779474210871641, 3.37497581786662752671429438514, 3.63509263252700718754121583203, 4.11717817004201581867908733121, 4.19207400774589290494702632423, 4.29948723498312921359661198126, 4.39015237267620016071973132426, 4.79026280665010030185415953282, 4.87475472329821282395741608023, 4.95636910060113215603955484263, 5.13453185508180892564053720610, 5.75929933302243275591941041872, 5.79541986940834833416421878507, 5.86180425310284841242924383360
