| L(s) = 1 | − 4·2-s + 2·4-s + 20·8-s − 45·16-s − 16·32-s − 12·49-s − 16·53-s − 16·59-s + 204·64-s + 16·67-s + 16·83-s − 48·89-s + 48·98-s + 48·101-s + 32·103-s + 64·106-s + 64·118-s − 12·121-s + 127-s − 232·128-s + 131-s − 64·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4-s + 7.07·8-s − 11.2·16-s − 2.82·32-s − 1.71·49-s − 2.19·53-s − 2.08·59-s + 51/2·64-s + 1.95·67-s + 1.75·83-s − 5.08·89-s + 4.84·98-s + 4.77·101-s + 3.15·103-s + 6.21·106-s + 5.89·118-s − 1.09·121-s + 0.0887·127-s − 20.5·128-s + 0.0873·131-s − 5.52·134-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3350015915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3350015915\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 5 | $C_4\times C_2$ | \( 1 + 48 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 + 12 T^{2} + 102 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 12 T^{2} + 150 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 12 T^{2} - 474 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 96 T^{2} + 3888 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 108 T^{2} + 4806 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 48 T^{2} + 1392 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 144 T^{2} + 8448 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 104 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 48 T^{2} + 7920 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 204 T^{2} + 18918 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 192 T^{2} + 18624 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 108 T^{2} + 13830 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 288 T^{2} + 37632 T^{4} + 288 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.47851006679254370843630862836, −5.93412208411902139274732195881, −5.91109262615766237941877015562, −5.85663006438876962859446808375, −5.38563819473272094834047403173, −5.06972689253680995906587776619, −4.90648667999292764884640040598, −4.89399306412213036626766665324, −4.79971642056925792004354375653, −4.29654631847430620736188859772, −4.28346123551140393892427555888, −4.07304138156508008277957963273, −3.89838175334764632466476978928, −3.47508501592036801709622007331, −3.17458327485022561324395143996, −3.01285051744317229979520709658, −2.99649124167832676873437049150, −1.91343348933356567842923015877, −1.90511377290118829918573787841, −1.87791294555356211329871204724, −1.68739395980479503988778820148, −1.07350905354962134852073729317, −0.805267071742919327659234690731, −0.47854134364188046178984894479, −0.38849266124131315326377670172,
0.38849266124131315326377670172, 0.47854134364188046178984894479, 0.805267071742919327659234690731, 1.07350905354962134852073729317, 1.68739395980479503988778820148, 1.87791294555356211329871204724, 1.90511377290118829918573787841, 1.91343348933356567842923015877, 2.99649124167832676873437049150, 3.01285051744317229979520709658, 3.17458327485022561324395143996, 3.47508501592036801709622007331, 3.89838175334764632466476978928, 4.07304138156508008277957963273, 4.28346123551140393892427555888, 4.29654631847430620736188859772, 4.79971642056925792004354375653, 4.89399306412213036626766665324, 4.90648667999292764884640040598, 5.06972689253680995906587776619, 5.38563819473272094834047403173, 5.85663006438876962859446808375, 5.91109262615766237941877015562, 5.93412208411902139274732195881, 6.47851006679254370843630862836
