| L(s) = 1 | + 4·3-s + 8·7-s + 8·9-s + 32·21-s + 12·27-s + 32·31-s − 32·37-s + 8·43-s + 32·49-s − 24·61-s + 64·63-s + 24·67-s − 32·73-s + 23·81-s + 128·93-s + 32·97-s + 24·103-s − 128·111-s − 20·121-s + 127-s + 32·129-s + 131-s + 137-s + 139-s + 128·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3.02·7-s + 8/3·9-s + 6.98·21-s + 2.30·27-s + 5.74·31-s − 5.26·37-s + 1.21·43-s + 32/7·49-s − 3.07·61-s + 8.06·63-s + 2.93·67-s − 3.74·73-s + 23/9·81-s + 13.2·93-s + 3.24·97-s + 2.36·103-s − 12.1·111-s − 1.81·121-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.5·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.01171362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.01171362\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 254 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 958 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 3682 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 3854 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 12878 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.81375719714500017868336676411, −7.54434022681102129259711095366, −7.27331507502432617979734143967, −7.25048871863829031643520472450, −6.99183340137787615935079542034, −6.43561408897137392191719354693, −6.32615977138206328877332237972, −6.12257249549774326151727324542, −5.77765085315488845289309526320, −5.27487929781584120190708097856, −4.93579949982330244563366460651, −4.87254526941122579339200741349, −4.81949693080983703297922330155, −4.47885609832175514594065076086, −4.22177179584801935656379819514, −3.85038367217096879344964794393, −3.42898663251707303857578393591, −3.34470924901840740743919297413, −2.86690011280739985368394662212, −2.57669095875863565541452278991, −2.43534831154197271873436270643, −1.85214976429759715755345088730, −1.75838974038615239535563466949, −1.31086555829133166954700244857, −0.936459502804013404569416563802,
0.936459502804013404569416563802, 1.31086555829133166954700244857, 1.75838974038615239535563466949, 1.85214976429759715755345088730, 2.43534831154197271873436270643, 2.57669095875863565541452278991, 2.86690011280739985368394662212, 3.34470924901840740743919297413, 3.42898663251707303857578393591, 3.85038367217096879344964794393, 4.22177179584801935656379819514, 4.47885609832175514594065076086, 4.81949693080983703297922330155, 4.87254526941122579339200741349, 4.93579949982330244563366460651, 5.27487929781584120190708097856, 5.77765085315488845289309526320, 6.12257249549774326151727324542, 6.32615977138206328877332237972, 6.43561408897137392191719354693, 6.99183340137787615935079542034, 7.25048871863829031643520472450, 7.27331507502432617979734143967, 7.54434022681102129259711095366, 7.81375719714500017868336676411
