| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 4·6-s + 4·7-s + 2·8-s + 9-s − 2·11-s + 2·12-s − 6·13-s − 8·14-s − 4·16-s + 4·17-s − 2·18-s − 4·19-s + 8·21-s + 4·22-s − 2·23-s + 4·24-s + 12·26-s − 2·27-s + 4·28-s + 4·29-s − 16·31-s + 2·32-s − 4·33-s − 8·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 1.63·6-s + 1.51·7-s + 0.707·8-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 1.66·13-s − 2.13·14-s − 16-s + 0.970·17-s − 0.471·18-s − 0.917·19-s + 1.74·21-s + 0.852·22-s − 0.417·23-s + 0.816·24-s + 2.35·26-s − 0.384·27-s + 0.755·28-s + 0.742·29-s − 2.87·31-s + 0.353·32-s − 0.696·33-s − 1.37·34-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
| \(L(1)\) |
\(\approx\) |
\(1.060221157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060221157\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 - 4 T + 8 T^{2} + 24 T^{3} - 97 T^{4} + 24 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 12 T^{2} + 24 T^{3} + 223 T^{4} + 24 p T^{5} - 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 16 T^{2} - 24 T^{3} + 359 T^{4} - 24 p T^{5} - 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 33 T^{2} - 18 T^{3} + 748 T^{4} - 18 p T^{5} - 33 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 6 T^{2} + 144 T^{3} - 821 T^{4} + 144 p T^{5} - 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 68 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 6 T - p T^{2} + 6 T^{3} + 2628 T^{4} + 6 p T^{5} - p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4 T - 60 T^{2} - 24 T^{3} + 3439 T^{4} - 24 p T^{5} - 60 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^3$ | \( 1 - 76 T^{2} + 3927 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 85 T^{2} - 6 T^{3} + 8724 T^{4} - 6 p T^{5} - 85 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 2 T - 49 T^{2} - 178 T^{3} - 2516 T^{4} - 178 p T^{5} - 49 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 84 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 4 T - 76 T^{2} + 344 T^{3} - 881 T^{4} + 344 p T^{5} - 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 151 T^{2} - 78 T^{3} + 14228 T^{4} - 78 p T^{5} - 151 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.68285294797155454505361840911, −6.25686207702355435639995207501, −6.15518262232401522170526628996, −6.10580767405119991570526807879, −5.47776282318177049317905358576, −5.33664182764822512454872074604, −5.32880741872113561338275996982, −5.08754041212455180151203209285, −4.85700934350983318777351197124, −4.55224326533151937581132194410, −4.32544095690904126012955567355, −4.20697970004621615522360582663, −3.96604971053456952418815318841, −3.59902653770573218864635646900, −3.35575679879851444754532898587, −2.91927798273618170141273003814, −2.84729153795899738508085833289, −2.72168741633367629780593088665, −2.30976443457264989185651218038, −1.77593064662545081790540316460, −1.75842472849520772257985959144, −1.60118504066294741333092796345, −1.41891922667167904604615989738, −0.47231651564855865962644614440, −0.33103585690432303766021295032,
0.33103585690432303766021295032, 0.47231651564855865962644614440, 1.41891922667167904604615989738, 1.60118504066294741333092796345, 1.75842472849520772257985959144, 1.77593064662545081790540316460, 2.30976443457264989185651218038, 2.72168741633367629780593088665, 2.84729153795899738508085833289, 2.91927798273618170141273003814, 3.35575679879851444754532898587, 3.59902653770573218864635646900, 3.96604971053456952418815318841, 4.20697970004621615522360582663, 4.32544095690904126012955567355, 4.55224326533151937581132194410, 4.85700934350983318777351197124, 5.08754041212455180151203209285, 5.32880741872113561338275996982, 5.33664182764822512454872074604, 5.47776282318177049317905358576, 6.10580767405119991570526807879, 6.15518262232401522170526628996, 6.25686207702355435639995207501, 6.68285294797155454505361840911
