| L(s) = 1 | + 8·7-s + 14·13-s + 32·19-s + 5·25-s − 58·31-s + 8·37-s + 14·43-s + 114·49-s − 124·61-s + 116·67-s − 208·73-s + 98·79-s + 112·91-s + 68·97-s − 196·103-s + 104·109-s − 197·121-s + 127-s + 131-s + 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 8/7·7-s + 1.07·13-s + 1.68·19-s + 1/5·25-s − 1.87·31-s + 8/37·37-s + 0.325·43-s + 2.32·49-s − 2.03·61-s + 1.73·67-s − 2.84·73-s + 1.24·79-s + 1.23·91-s + 0.701·97-s − 1.90·103-s + 0.954·109-s − 1.62·121-s + 0.00787·127-s + 0.00763·131-s + 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.059850666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.059850666\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T - 33 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 197 T^{2} + 24168 T^{4} + 197 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 7 T - 120 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 173 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 1013 T^{2} + 746328 T^{4} + 1013 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 523 T^{2} - 433752 T^{4} - 523 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 29 T - 120 T^{2} + 29 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 3182 T^{2} + 7299363 T^{4} + 3182 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 7 T - 1800 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 3293 T^{2} + 5964168 T^{4} + 3293 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 3202 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 5342 T^{2} + 16419603 T^{4} + 5342 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 62 T + 123 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 58 T - 1125 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7202 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 52 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 49 T - 3840 T^{2} - 49 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 13598 T^{2} + 137447283 T^{4} + 13598 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2}( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 34 T - 8253 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.55085647517169559684226120151, −6.22901091927213492414315077105, −6.04180675168862028860488551250, −5.85568604005676540115321117120, −5.53564619778701109705260881929, −5.34798207824931710943898732225, −5.22511620723386294947314992134, −5.10885599121157546219717218301, −4.90329751072685881754224048485, −4.30802335650193782142246720431, −4.23012246332210650562020163787, −4.19438154941279917533481281577, −3.75326914820335103540961405788, −3.70227473342916501889284490951, −3.26217676047341639371303843804, −3.05830996482848064030831375529, −2.83315268829907000053262748216, −2.53993188217001000921961169736, −2.19869979302668064084692557742, −1.84346147574778854551574914282, −1.60657944360340906418156716314, −1.29821439364242329130495703150, −1.06674401821678764807363768299, −0.77611087880107809444988799286, −0.16564842436604128501098413751,
0.16564842436604128501098413751, 0.77611087880107809444988799286, 1.06674401821678764807363768299, 1.29821439364242329130495703150, 1.60657944360340906418156716314, 1.84346147574778854551574914282, 2.19869979302668064084692557742, 2.53993188217001000921961169736, 2.83315268829907000053262748216, 3.05830996482848064030831375529, 3.26217676047341639371303843804, 3.70227473342916501889284490951, 3.75326914820335103540961405788, 4.19438154941279917533481281577, 4.23012246332210650562020163787, 4.30802335650193782142246720431, 4.90329751072685881754224048485, 5.10885599121157546219717218301, 5.22511620723386294947314992134, 5.34798207824931710943898732225, 5.53564619778701109705260881929, 5.85568604005676540115321117120, 6.04180675168862028860488551250, 6.22901091927213492414315077105, 6.55085647517169559684226120151
