| L(s) = 1 | − 12·11-s + 4·13-s + 24·23-s − 2·25-s + 8·37-s − 24·47-s + 4·49-s − 12·59-s − 4·61-s − 12·71-s + 4·73-s − 32·97-s + 48·107-s + 4·109-s + 52·121-s + 127-s + 131-s + 137-s + 139-s − 48·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + ⋯ |
| L(s) = 1 | − 3.61·11-s + 1.10·13-s + 5.00·23-s − 2/5·25-s + 1.31·37-s − 3.50·47-s + 4/7·49-s − 1.56·59-s − 0.512·61-s − 1.42·71-s + 0.468·73-s − 3.24·97-s + 4.64·107-s + 0.383·109-s + 4.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.01·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-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
| \(L(1)\) |
\(\approx\) |
\(2.843514621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.843514621\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 579 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 12 T + 79 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 1194 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 82 T^{2} + 3171 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4506 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 1002 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 86 T^{2} + 3579 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 202 T^{2} + 20955 T^{4} - 202 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 139 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11994 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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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
−6.54801771529231945686448002656, −6.21719957552703635874923542516, −5.88089786954278546467640145991, −5.87922755984119348496344402269, −5.53847089177282888754189162777, −5.37626723346144320415770465442, −5.13870464228729663608012801838, −4.97232893074457751734717892370, −4.72271820736228343799453897253, −4.69909309616188821483787809321, −4.58802398679959866710061836122, −4.07932918198831638818367437533, −3.84678764616181897334961248691, −3.28590817834723512813050154802, −3.24554581397598036876368584785, −3.18986345766012403224163924120, −2.83995562186468823654378980564, −2.67201338497142055084619081595, −2.62688782480522439807356017849, −2.03031818208954932468058188193, −1.80484765349897089130126594125, −1.31398808466998804631050492899, −1.23032000481983733540545183292, −0.52576368104977398623015708558, −0.43631070961811189632071913433,
0.43631070961811189632071913433, 0.52576368104977398623015708558, 1.23032000481983733540545183292, 1.31398808466998804631050492899, 1.80484765349897089130126594125, 2.03031818208954932468058188193, 2.62688782480522439807356017849, 2.67201338497142055084619081595, 2.83995562186468823654378980564, 3.18986345766012403224163924120, 3.24554581397598036876368584785, 3.28590817834723512813050154802, 3.84678764616181897334961248691, 4.07932918198831638818367437533, 4.58802398679959866710061836122, 4.69909309616188821483787809321, 4.72271820736228343799453897253, 4.97232893074457751734717892370, 5.13870464228729663608012801838, 5.37626723346144320415770465442, 5.53847089177282888754189162777, 5.87922755984119348496344402269, 5.88089786954278546467640145991, 6.21719957552703635874923542516, 6.54801771529231945686448002656
