| L(s) = 1 | + 4·2-s + 4·4-s − 6·5-s − 16·8-s − 24·10-s − 10·13-s − 64·16-s − 44·17-s − 24·20-s + 25·25-s − 40·26-s − 120·29-s − 64·32-s − 176·34-s − 72·37-s + 96·40-s + 142·41-s + 98·49-s + 100·50-s − 40·52-s + 146·53-s − 480·58-s + 120·61-s + 192·64-s + 60·65-s − 176·68-s − 192·73-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s − 6/5·5-s − 2·8-s − 2.39·10-s − 0.769·13-s − 4·16-s − 2.58·17-s − 6/5·20-s + 25-s − 1.53·26-s − 4.13·29-s − 2·32-s − 5.17·34-s − 1.94·37-s + 12/5·40-s + 3.46·41-s + 2·49-s + 2·50-s − 0.769·52-s + 2.75·53-s − 8.27·58-s + 1.96·61-s + 3·64-s + 0.923·65-s − 2.58·68-s − 2.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.08124002377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08124002377\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 11 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 10 T - 69 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2}( 1 - 16 T - 33 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 19 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2}( 1 + 40 T + 759 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2}( 1 + 24 T - 793 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2}( 1 + 18 T - 1357 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 43 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 90 T + 5291 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} )( 1 - 56 T + 327 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 59 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{2}( 1 + 120 T + 10679 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 96 T + 3887 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 160 T + 17679 T^{2} - 160 p^{2} T^{3} + p^{4} T^{4} )( 1 + 78 T - 1837 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 97 | $C_2^2$ | \( ( 1 + 130 T + 7491 T^{2} + 130 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.529152062621353120095155258715, −8.464396109754202886430214713705, −7.82992852958965779912071731549, −7.40803816258959959825654599460, −7.37079490124274250383962344425, −7.19269476042859689844140971914, −7.08842166769557471044944751827, −6.43270685787262965849293546315, −6.23061487473746780763614373655, −5.94786019339252064760432167952, −5.74885026234471136362184109955, −5.36248750551441299484118433192, −5.09012753925589410749873489176, −4.92184439290780085244027675646, −4.55578696641645476921592095116, −4.04652257857251099590111998408, −3.99792330443451633356024890771, −3.89589931970024439302200029422, −3.61765338255912767087677587517, −3.11928409264883454019076794116, −2.42369428265830234690704845769, −2.33305663324311476801215997861, −2.16409893359054228072597409828, −0.857636742760764799140038248898, −0.06308909942465457143872916466,
0.06308909942465457143872916466, 0.857636742760764799140038248898, 2.16409893359054228072597409828, 2.33305663324311476801215997861, 2.42369428265830234690704845769, 3.11928409264883454019076794116, 3.61765338255912767087677587517, 3.89589931970024439302200029422, 3.99792330443451633356024890771, 4.04652257857251099590111998408, 4.55578696641645476921592095116, 4.92184439290780085244027675646, 5.09012753925589410749873489176, 5.36248750551441299484118433192, 5.74885026234471136362184109955, 5.94786019339252064760432167952, 6.23061487473746780763614373655, 6.43270685787262965849293546315, 7.08842166769557471044944751827, 7.19269476042859689844140971914, 7.37079490124274250383962344425, 7.40803816258959959825654599460, 7.82992852958965779912071731549, 8.464396109754202886430214713705, 8.529152062621353120095155258715
