| L(s) = 1 | − 2·2-s + 4-s − 6·7-s + 2·8-s + 9-s + 6·11-s − 6·13-s + 12·14-s − 4·16-s + 18·17-s − 2·18-s + 6·19-s − 12·22-s − 18·23-s + 12·26-s − 6·28-s − 6·29-s + 2·32-s − 36·34-s + 36-s − 6·37-s − 12·38-s + 12·41-s + 18·43-s + 6·44-s + 36·46-s − 12·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 2.26·7-s + 0.707·8-s + 1/3·9-s + 1.80·11-s − 1.66·13-s + 3.20·14-s − 16-s + 4.36·17-s − 0.471·18-s + 1.37·19-s − 2.55·22-s − 3.75·23-s + 2.35·26-s − 1.13·28-s − 1.11·29-s + 0.353·32-s − 6.17·34-s + 1/6·36-s − 0.986·37-s − 1.94·38-s + 1.87·41-s + 2.74·43-s + 0.904·44-s + 5.30·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.633037829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.633037829\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 6 T + 16 T^{2} + 36 T^{3} + 99 T^{4} + 36 p T^{5} + 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 28 T^{2} - 96 T^{3} + 267 T^{4} - 96 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 18 T + 172 T^{2} + 1152 T^{3} + 6051 T^{4} + 1152 p T^{5} + 172 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 133 T^{2} - 1020 T^{3} + 7512 T^{4} - 1020 p T^{5} + 133 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 18 T + 220 T^{2} - 2016 T^{3} + 15339 T^{4} - 2016 p T^{5} + 220 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 24 T + 251 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 205 T^{2} + 1460 T^{3} + 9904 T^{4} + 1460 p T^{5} + 205 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 112 T^{2} - 1404 T^{3} + 18747 T^{4} - 1404 p T^{5} + 112 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18 T + 268 T^{2} - 2880 T^{3} + 28227 T^{4} - 2880 p T^{5} + 268 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.69025809877059780457286066386, −6.41384542843255476010328739160, −5.96311589970546707830404867324, −5.96220329721354660290609586525, −5.81532075590722478644480351391, −5.54174790402080838891615237606, −5.31742725097340975405348848031, −5.18337641443156932935946125744, −5.09437142136594132410955154994, −4.27479349550490764743172932794, −4.22674022738484233140536031448, −4.01851839710187478518920441036, −3.99937605032833357132516089858, −3.56821306658274984288432855440, −3.33767152909756901003176324416, −3.22484694767064601731363355317, −3.19952996331198157130293456876, −2.36053324898845470639438050208, −2.35020759919414857570872839979, −2.14079282762072907292109777254, −1.70598090187077076039991618599, −1.27278625443962318852434189435, −0.899895902825374714813872590449, −0.65257488186604479019493813770, −0.48855867159325886538671052192,
0.48855867159325886538671052192, 0.65257488186604479019493813770, 0.899895902825374714813872590449, 1.27278625443962318852434189435, 1.70598090187077076039991618599, 2.14079282762072907292109777254, 2.35020759919414857570872839979, 2.36053324898845470639438050208, 3.19952996331198157130293456876, 3.22484694767064601731363355317, 3.33767152909756901003176324416, 3.56821306658274984288432855440, 3.99937605032833357132516089858, 4.01851839710187478518920441036, 4.22674022738484233140536031448, 4.27479349550490764743172932794, 5.09437142136594132410955154994, 5.18337641443156932935946125744, 5.31742725097340975405348848031, 5.54174790402080838891615237606, 5.81532075590722478644480351391, 5.96220329721354660290609586525, 5.96311589970546707830404867324, 6.41384542843255476010328739160, 6.69025809877059780457286066386
