| L(s) = 1 | − 2·5-s + 7-s + 3·11-s + 2·13-s + 18·17-s − 2·19-s + 3·23-s + 25-s − 3·29-s − 2·31-s − 2·35-s − 16·37-s − 6·41-s − 17·43-s + 9·47-s + 6·49-s − 6·55-s − 3·59-s − 61-s − 4·65-s − 14·67-s − 24·71-s − 22·73-s + 3·77-s + 4·79-s − 9·83-s − 36·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s + 4.36·17-s − 0.458·19-s + 0.625·23-s + 1/5·25-s − 0.557·29-s − 0.359·31-s − 0.338·35-s − 2.63·37-s − 0.937·41-s − 2.59·43-s + 1.31·47-s + 6/7·49-s − 0.809·55-s − 0.390·59-s − 0.128·61-s − 0.496·65-s − 1.71·67-s − 2.84·71-s − 2.57·73-s + 0.341·77-s + 0.450·79-s − 0.987·83-s − 3.90·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3913165071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3913165071\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - T - 5 T^{2} + 8 T^{3} - 20 T^{4} + 8 p T^{5} - 5 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2 T + 10 T^{2} + 64 T^{3} - 185 T^{4} + 64 p T^{5} + 10 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 T - 31 T^{2} + 18 T^{3} + 864 T^{4} + 18 p T^{5} - 31 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 + 2 T - 26 T^{2} - 64 T^{3} - 185 T^{4} - 64 p T^{5} - 26 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 17 T + 139 T^{2} + 1088 T^{3} + 8224 T^{4} + 1088 p T^{5} + 139 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 108 p T^{5} - 25 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 3 T - 103 T^{2} - 18 T^{3} + 8532 T^{4} - 18 p T^{5} - 103 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 74 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 + 11 T + 102 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 9 T - 97 T^{2} + 108 T^{3} + 18072 T^{4} + 108 p T^{5} - 97 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 15 T + 160 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 11 T - 95 T^{2} - 242 T^{3} + 25510 T^{4} - 242 p T^{5} - 95 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.60719439416172647058980097680, −6.04861493998483332078206496686, −5.84445433871508404300560822672, −5.84315399154088439119580258964, −5.57894415115371518130931005317, −5.49055643837319231914326888234, −5.41579970014688605350626889843, −4.77664744588682740242015216761, −4.75876216426576077822711256841, −4.61386431427759819082598315660, −4.32545865104193855440291381480, −4.01660146687051783073530087133, −3.71856314478104067363303091762, −3.48297232477116297028425260396, −3.40561046180895867081244788786, −3.33554607798702019121595924936, −2.97106047633670901965720122850, −2.82065025277579673759799089896, −2.40724324228254774161873856449, −1.68980191857602202439598792166, −1.65818213202416626105002197996, −1.49284249398105826957327862181, −1.23443358669135368815778335351, −0.867548070377326672663479227381, −0.10535307833362608397612487114,
0.10535307833362608397612487114, 0.867548070377326672663479227381, 1.23443358669135368815778335351, 1.49284249398105826957327862181, 1.65818213202416626105002197996, 1.68980191857602202439598792166, 2.40724324228254774161873856449, 2.82065025277579673759799089896, 2.97106047633670901965720122850, 3.33554607798702019121595924936, 3.40561046180895867081244788786, 3.48297232477116297028425260396, 3.71856314478104067363303091762, 4.01660146687051783073530087133, 4.32545865104193855440291381480, 4.61386431427759819082598315660, 4.75876216426576077822711256841, 4.77664744588682740242015216761, 5.41579970014688605350626889843, 5.49055643837319231914326888234, 5.57894415115371518130931005317, 5.84315399154088439119580258964, 5.84445433871508404300560822672, 6.04861493998483332078206496686, 6.60719439416172647058980097680
