| L(s) = 1 | − 12·7-s − 16-s − 12·19-s + 12·31-s − 24·37-s + 72·49-s + 48·61-s − 12·67-s + 24·73-s + 24·97-s − 48·109-s + 12·112-s + 127-s + 131-s + 144·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4.53·7-s − 1/4·16-s − 2.75·19-s + 2.15·31-s − 3.94·37-s + 72/7·49-s + 6.14·61-s − 1.46·67-s + 2.80·73-s + 2.43·97-s − 4.59·109-s + 1.13·112-s + 0.0887·127-s + 0.0873·131-s + 12.4·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07548128508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07548128508\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + T^{4} + p^{4} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )( 1 + 80 T^{2} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 4382 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 3442 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - 3998 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 + 14434 T^{4} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.72666759651136570460989858431, −6.51109058107440059983541423036, −6.28703502835535769660325346499, −6.26929236156448205072880694937, −6.26206686951174409734975596025, −5.62372060149902290761014163543, −5.50040015797191600691068796728, −5.32656030980636682488613365273, −5.12515055575866803544549903382, −4.79091960304865898689813839375, −4.27909345101776899700686401541, −4.19339367881725160998816195317, −4.06898533412037361733665441412, −3.61981259152175012679301830676, −3.50319093855173613366747266344, −3.39961941439164172616721412079, −3.20151812018879670900117928078, −2.71675771177183049362329090954, −2.57121782399025988743241781438, −2.26242994136238891479389358776, −2.14797900785900019869816493927, −1.69005368256204279068614945095, −0.905244665536004335488998220099, −0.60937875118414608073406066828, −0.087040092856759714423563757227,
0.087040092856759714423563757227, 0.60937875118414608073406066828, 0.905244665536004335488998220099, 1.69005368256204279068614945095, 2.14797900785900019869816493927, 2.26242994136238891479389358776, 2.57121782399025988743241781438, 2.71675771177183049362329090954, 3.20151812018879670900117928078, 3.39961941439164172616721412079, 3.50319093855173613366747266344, 3.61981259152175012679301830676, 4.06898533412037361733665441412, 4.19339367881725160998816195317, 4.27909345101776899700686401541, 4.79091960304865898689813839375, 5.12515055575866803544549903382, 5.32656030980636682488613365273, 5.50040015797191600691068796728, 5.62372060149902290761014163543, 6.26206686951174409734975596025, 6.26929236156448205072880694937, 6.28703502835535769660325346499, 6.51109058107440059983541423036, 6.72666759651136570460989858431
