| L(s) = 1 | − 3-s + 5-s + 3·7-s + 3·9-s + 2·11-s − 5·13-s − 15-s − 10·17-s − 14·19-s − 3·21-s + 5·23-s + 2·25-s − 8·27-s + 3·29-s + 7·31-s − 2·33-s + 3·35-s + 12·37-s + 5·39-s + 12·41-s + 8·43-s + 3·45-s + 3·47-s + 8·49-s + 10·51-s − 20·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.13·7-s + 9-s + 0.603·11-s − 1.38·13-s − 0.258·15-s − 2.42·17-s − 3.21·19-s − 0.654·21-s + 1.04·23-s + 2/5·25-s − 1.53·27-s + 0.557·29-s + 1.25·31-s − 0.348·33-s + 0.507·35-s + 1.97·37-s + 0.800·39-s + 1.87·41-s + 1.21·43-s + 0.447·45-s + 0.437·47-s + 8/7·49-s + 1.40·51-s − 2.74·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.163643027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163643027\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 8 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3 T + T^{2} + 18 T^{3} - 48 T^{4} + 18 p T^{5} + p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 10 p T^{5} + p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 5 T - 19 T^{2} + 10 T^{3} + 832 T^{4} + 10 p T^{5} - 19 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 7 T - 17 T^{2} - 28 T^{3} + 1876 T^{4} - 28 p T^{5} - 17 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 36 p T^{5} + 59 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T - 5 T^{2} + 136 T^{3} + 160 T^{4} + 136 p T^{5} - 5 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3 T - 79 T^{2} + 18 T^{3} + 5112 T^{4} + 18 p T^{5} - 79 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 8 p T^{5} - 113 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 89 T^{2} + 116 T^{3} + 5464 T^{4} + 116 p T^{5} - 89 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7 T - 113 T^{2} + 28 T^{3} + 16132 T^{4} + 28 p T^{5} - 113 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 25 T + 311 T^{2} - 3700 T^{3} + 39832 T^{4} - 3700 p T^{5} + 311 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 136 p T^{5} - 113 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.719821309270489066472683815633, −9.168641104253105348380164417819, −9.163271679503416627936204475637, −8.925412211009464225880284705913, −8.653446361384281559567406176400, −8.207484316760569130376985957768, −7.888672098459935273545477497692, −7.65962270926953171518328201763, −7.54463706694837679097357205459, −6.75724365807925030726286782373, −6.68090154786800399856727105782, −6.61774342285737836548211552451, −6.35068227173636875972482030512, −5.71682413458066262014292123890, −5.71071836576077775830850192547, −4.96380235069853970123839412052, −4.67165229832493134016596280304, −4.44347302238199323016451924215, −4.26901520700210667846244787695, −4.20050406750091340005308759601, −3.28763326427326427239363895032, −2.32745663651917742497329581627, −2.30975243910824142642966673921, −2.11686393901895015657619816444, −0.945217057579748622313714488255,
0.945217057579748622313714488255, 2.11686393901895015657619816444, 2.30975243910824142642966673921, 2.32745663651917742497329581627, 3.28763326427326427239363895032, 4.20050406750091340005308759601, 4.26901520700210667846244787695, 4.44347302238199323016451924215, 4.67165229832493134016596280304, 4.96380235069853970123839412052, 5.71071836576077775830850192547, 5.71682413458066262014292123890, 6.35068227173636875972482030512, 6.61774342285737836548211552451, 6.68090154786800399856727105782, 6.75724365807925030726286782373, 7.54463706694837679097357205459, 7.65962270926953171518328201763, 7.888672098459935273545477497692, 8.207484316760569130376985957768, 8.653446361384281559567406176400, 8.925412211009464225880284705913, 9.163271679503416627936204475637, 9.168641104253105348380164417819, 9.719821309270489066472683815633
