| L(s) = 1 | − 2·11-s − 8·13-s − 12·23-s + 3·25-s − 20·37-s − 20·47-s − 49-s + 8·59-s + 16·71-s − 6·73-s − 18·83-s + 14·97-s − 34·107-s − 4·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 16·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 2.21·13-s − 2.50·23-s + 3/5·25-s − 3.28·37-s − 2.91·47-s − 1/7·49-s + 1.04·59-s + 1.89·71-s − 0.702·73-s − 1.97·83-s + 1.42·97-s − 3.28·107-s − 0.383·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.33·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 93 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−9.249179124196987575960850906124, −8.550660249015272561441967271041, −8.242268402615046452556269929050, −8.110799401224723554800218536078, −7.49457376896425652751747497860, −7.10767312475555086792362137668, −6.87123009384917670787129941289, −6.36418245293273840739389269451, −5.87467154560491979957505767945, −5.25852477156993402855956031442, −4.96942438208372425720561034500, −4.87886908545961866961012833648, −3.91889873088688425934273734374, −3.78573298617316212330687321750, −2.98790367594008837561174575716, −2.57331871558981163156946589221, −1.98390570863539637619059405532, −1.58214941398350325894069136048, 0, 0,
1.58214941398350325894069136048, 1.98390570863539637619059405532, 2.57331871558981163156946589221, 2.98790367594008837561174575716, 3.78573298617316212330687321750, 3.91889873088688425934273734374, 4.87886908545961866961012833648, 4.96942438208372425720561034500, 5.25852477156993402855956031442, 5.87467154560491979957505767945, 6.36418245293273840739389269451, 6.87123009384917670787129941289, 7.10767312475555086792362137668, 7.49457376896425652751747497860, 8.110799401224723554800218536078, 8.242268402615046452556269929050, 8.550660249015272561441967271041, 9.249179124196987575960850906124
