| L(s) = 1 | − 16·11-s + 4·16-s − 4·19-s + 8·29-s − 12·31-s − 8·41-s + 4·49-s + 40·59-s + 16·61-s − 8·71-s − 48·79-s + 24·101-s + 52·109-s + 122·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s − 64·176-s + 179-s + ⋯ |
| L(s) = 1 | − 4.82·11-s + 16-s − 0.917·19-s + 1.48·29-s − 2.15·31-s − 1.24·41-s + 4/7·49-s + 5.20·59-s + 2.04·61-s − 0.949·71-s − 5.40·79-s + 2.38·101-s + 4.98·109-s + 11.0·121-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 + 1.53·169-s + 0.0760·173-s − 4.82·176-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.211366624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.211366624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1466 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 7626 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2154 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 5594 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 11402 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 20 T + 215 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 143 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 15258 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 29706 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 32442 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.45877213061806128903308929115, −6.15788297013735382563986648169, −6.08267121493940639388107077075, −5.65639207195702036337256333143, −5.57301587193185918398847097280, −5.46241090228724010396829275500, −5.18921325623164083039062317798, −5.12309486758904181405329137113, −5.07052352779606431056583633597, −4.54391002278571785271266190062, −4.49297902159190795037728037390, −3.94269265691319818706658022766, −3.93731722232933425983024296761, −3.79389246189839866561908683237, −3.19988725232541530320876935937, −2.96914950105314895806336775733, −2.87209326532031997101080514258, −2.78937284844986728808227080122, −2.44722476970710319474752121084, −1.95187557900100299219476778401, −1.87828592089834310639286359870, −1.86519193110810062509929848644, −0.818582328824752729143874248231, −0.67932172919178077325894210413, −0.27245872938044513129069274538,
0.27245872938044513129069274538, 0.67932172919178077325894210413, 0.818582328824752729143874248231, 1.86519193110810062509929848644, 1.87828592089834310639286359870, 1.95187557900100299219476778401, 2.44722476970710319474752121084, 2.78937284844986728808227080122, 2.87209326532031997101080514258, 2.96914950105314895806336775733, 3.19988725232541530320876935937, 3.79389246189839866561908683237, 3.93731722232933425983024296761, 3.94269265691319818706658022766, 4.49297902159190795037728037390, 4.54391002278571785271266190062, 5.07052352779606431056583633597, 5.12309486758904181405329137113, 5.18921325623164083039062317798, 5.46241090228724010396829275500, 5.57301587193185918398847097280, 5.65639207195702036337256333143, 6.08267121493940639388107077075, 6.15788297013735382563986648169, 6.45877213061806128903308929115
