| L(s) = 1 | − 2·4-s + 3·9-s + 16·11-s + 3·16-s + 4·19-s − 2·29-s − 4·31-s − 6·36-s + 16·41-s − 32·44-s + 19·49-s − 2·59-s + 28·61-s − 4·64-s + 8·71-s − 8·76-s + 4·79-s − 7·81-s − 8·89-s + 48·99-s − 20·101-s − 6·109-s + 4·116-s + 116·121-s + 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s + 9-s + 4.82·11-s + 3/4·16-s + 0.917·19-s − 0.371·29-s − 0.718·31-s − 36-s + 2.49·41-s − 4.82·44-s + 19/7·49-s − 0.260·59-s + 3.58·61-s − 1/2·64-s + 0.949·71-s − 0.917·76-s + 0.450·79-s − 7/9·81-s − 0.847·89-s + 4.82·99-s − 1.99·101-s − 0.574·109-s + 0.371·116-s + 10.5·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.866725059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.866725059\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 19 T^{2} + 184 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + T^{2} + 64 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 11 T^{2} + 744 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 766 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 3814 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 191 T^{2} + 14632 T^{4} - 191 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + T + 114 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 259 T^{2} + 25744 T^{4} - 259 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 175 T^{2} + 15216 T^{4} - 175 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 20446 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.30443899570842720861496848167, −6.84959769768094511393835319048, −6.79935286416154533119717229410, −6.55742262621085631138954296499, −6.35918334607721390455905119980, −6.11249815152604386719311884380, −5.83488378895036279430932122721, −5.63707008418649172748193183955, −5.31571301029667086421113551717, −5.17277391746901531548056398662, −4.85734995906847119650561445981, −4.35049974592388618370427717040, −4.22856607826885866796842447461, −4.00275482205706600790667209027, −3.89208959887610682249731555889, −3.85636372636566570484734243734, −3.66794297247418809718432623874, −3.06319225126066391878135551083, −2.81206294397502009932319731852, −2.33354802587396489548089853544, −1.95037387115377977820933789498, −1.54407781671307853647558981084, −1.26130400033214096309090526784, −0.887244662693748130659177322112, −0.817839717350066967349898810547,
0.817839717350066967349898810547, 0.887244662693748130659177322112, 1.26130400033214096309090526784, 1.54407781671307853647558981084, 1.95037387115377977820933789498, 2.33354802587396489548089853544, 2.81206294397502009932319731852, 3.06319225126066391878135551083, 3.66794297247418809718432623874, 3.85636372636566570484734243734, 3.89208959887610682249731555889, 4.00275482205706600790667209027, 4.22856607826885866796842447461, 4.35049974592388618370427717040, 4.85734995906847119650561445981, 5.17277391746901531548056398662, 5.31571301029667086421113551717, 5.63707008418649172748193183955, 5.83488378895036279430932122721, 6.11249815152604386719311884380, 6.35918334607721390455905119980, 6.55742262621085631138954296499, 6.79935286416154533119717229410, 6.84959769768094511393835319048, 7.30443899570842720861496848167
