| L(s) = 1 | + 8·4-s − 3.27e4·16-s − 5.96e4·19-s − 9.76e5·31-s − 1.70e6·49-s + 7.04e6·61-s − 3.92e5·64-s − 4.77e5·76-s − 8.67e6·79-s − 6.81e7·109-s − 4.79e7·121-s − 7.81e6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 5.84e7·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1/16·4-s − 1.99·16-s − 1.99·19-s − 5.88·31-s − 2.06·49-s + 3.97·61-s − 0.187·64-s − 0.124·76-s − 1.97·79-s − 5.03·109-s − 2.46·121-s − 0.368·124-s − 0.931·169-s − 0.129·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{14} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 851486 T^{2} + p^{14} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 23980342 T^{2} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 2248418 p T^{2} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 701511806 T^{2} + p^{14} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 14924 T + p^{7} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 6809599934 T^{2} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 15830998618 T^{2} + p^{14} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 244192 T + p^{7} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 71354798134 T^{2} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 9763771762 T^{2} + p^{14} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 112663884614 T^{2} + p^{14} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 731839822834 T^{2} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1126336371734 T^{2} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 1127453015638 T^{2} + p^{14} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 1762442 T + p^{7} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 11134086793046 T^{2} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 17983481012782 T^{2} + p^{14} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 5638364775794 T^{2} + p^{14} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2168216 T + p^{7} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 53712042805814 T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 60261709791058 T^{2} + p^{14} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 33799121370626 T^{2} + p^{14} T^{4} )^{2} \) |
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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
−8.306838908018685679993823998708, −7.72391867239091483256009633613, −7.67756694723461061341555446538, −7.29333157501200430294927903931, −6.94798751965806917001964616053, −6.86309178915583516108256014769, −6.74192027592002173090294436582, −6.29857300765651462634515864815, −6.06859147352436127819581930975, −5.60773815824705763942100413123, −5.35902985944957155376666303860, −5.26294594477942871336462368973, −4.94387805871441033615300457861, −4.49892344038861788032173049376, −4.11692747473427442533360164658, −3.87090848160818030175201925476, −3.86971812263792737135117133982, −3.42314368365558938736199382548, −2.97414286160303604333852679068, −2.49366241697531983146325256947, −2.22434712183141932618072026675, −1.99467393098629891695443157315, −1.85062384447641692865375323474, −1.28304603539879074633043361601, −1.06722626817302333872703462501, 0, 0, 0, 0,
1.06722626817302333872703462501, 1.28304603539879074633043361601, 1.85062384447641692865375323474, 1.99467393098629891695443157315, 2.22434712183141932618072026675, 2.49366241697531983146325256947, 2.97414286160303604333852679068, 3.42314368365558938736199382548, 3.86971812263792737135117133982, 3.87090848160818030175201925476, 4.11692747473427442533360164658, 4.49892344038861788032173049376, 4.94387805871441033615300457861, 5.26294594477942871336462368973, 5.35902985944957155376666303860, 5.60773815824705763942100413123, 6.06859147352436127819581930975, 6.29857300765651462634515864815, 6.74192027592002173090294436582, 6.86309178915583516108256014769, 6.94798751965806917001964616053, 7.29333157501200430294927903931, 7.67756694723461061341555446538, 7.72391867239091483256009633613, 8.306838908018685679993823998708
