L(s) = 1 | − 2·4-s − 2·7-s − 4·9-s − 4·11-s + 8·13-s − 2·17-s + 2·19-s − 12·23-s − 8·25-s + 4·28-s − 8·29-s + 4·31-s + 8·36-s − 14·37-s + 16·41-s + 8·44-s + 8·47-s − 11·49-s − 16·52-s − 4·53-s − 12·59-s + 20·61-s + 8·63-s + 8·64-s + 14·67-s + 4·68-s + 12·71-s + ⋯ |
L(s) = 1 | − 4-s − 0.755·7-s − 4/3·9-s − 1.20·11-s + 2.21·13-s − 0.485·17-s + 0.458·19-s − 2.50·23-s − 8/5·25-s + 0.755·28-s − 1.48·29-s + 0.718·31-s + 4/3·36-s − 2.30·37-s + 2.49·41-s + 1.20·44-s + 1.16·47-s − 1.57·49-s − 2.21·52-s − 0.549·53-s − 1.56·59-s + 2.56·61-s + 1.00·63-s + 64-s + 1.71·67-s + 0.485·68-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16024009 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16024009 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 4003 | | \( 1+O(T) \) |
good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 60 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 151 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 251 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 212 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.197238268154793147296299303082, −8.151364239265583733926626653248, −7.63008765758379215224417701137, −7.38015678836174200313665146198, −6.55663777766642901951761733324, −6.26011756104634778170814493927, −6.07297894844572667204275783056, −5.70019151937449400557254896563, −5.26053723403507263980916319769, −5.13954118223872036146552147011, −4.23832529337291424040604495150, −4.01290259266816417020876944909, −3.56605559268132397838855381812, −3.50585943082748260518053743449, −2.71798638211430985859676953885, −2.18466034537402874734904992469, −1.90092769902051903625201955119, −0.898079468955872573049789736613, 0, 0,
0.898079468955872573049789736613, 1.90092769902051903625201955119, 2.18466034537402874734904992469, 2.71798638211430985859676953885, 3.50585943082748260518053743449, 3.56605559268132397838855381812, 4.01290259266816417020876944909, 4.23832529337291424040604495150, 5.13954118223872036146552147011, 5.26053723403507263980916319769, 5.70019151937449400557254896563, 6.07297894844572667204275783056, 6.26011756104634778170814493927, 6.55663777766642901951761733324, 7.38015678836174200313665146198, 7.63008765758379215224417701137, 8.151364239265583733926626653248, 8.197238268154793147296299303082