| L(s) = 1 | − 2·2-s + 4-s − 2·7-s + 2·8-s + 9-s − 6·11-s − 10·13-s + 4·14-s − 4·16-s − 6·17-s − 2·18-s − 6·19-s + 12·22-s + 18·23-s + 20·26-s − 2·28-s − 2·29-s + 2·32-s + 12·34-s + 36-s + 14·37-s + 12·38-s + 36·41-s + 30·43-s − 6·44-s − 36·46-s + 12·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.755·7-s + 0.707·8-s + 1/3·9-s − 1.80·11-s − 2.77·13-s + 1.06·14-s − 16-s − 1.45·17-s − 0.471·18-s − 1.37·19-s + 2.55·22-s + 3.75·23-s + 3.92·26-s − 0.377·28-s − 0.371·29-s + 0.353·32-s + 2.05·34-s + 1/6·36-s + 2.30·37-s + 1.94·38-s + 5.62·41-s + 4.57·43-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
| \(L(1)\) |
\(\approx\) |
\(1.179576191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.179576191\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 2 T - 8 T^{2} - 4 T^{3} + 67 T^{4} - 4 p T^{5} - 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 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$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 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 | $D_4\times C_2$ | \( 1 - 18 T + 180 T^{2} - 1296 T^{3} + 7139 T^{4} - 1296 p T^{5} + 180 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 43 T^{2} - 22 T^{3} + 1252 T^{4} - 22 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 14 T + 85 T^{2} - 14 p T^{3} + 100 p T^{4} - 14 p^{2} T^{5} + 85 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 36 T + 621 T^{2} - 6804 T^{3} + 51752 T^{4} - 6804 p T^{5} + 621 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 30 T + 460 T^{2} - 4800 T^{3} + 36651 T^{4} - 4800 p T^{5} + 460 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 170 T^{2} + 12411 T^{4} - 170 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 2 T + 16 T^{2} - 292 T^{3} - 4613 T^{4} - 292 p T^{5} + 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 148 T^{2} - 816 T^{3} + 14307 T^{4} - 816 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 207 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 10 T + 164 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−6.60811824940103760070520816662, −6.33524267824542268484464267993, −5.98890025400125078948367871776, −5.91083659788404491074606441399, −5.81582228857043371436225142245, −5.53899519571204408873603913648, −5.40182182218295441464243685702, −4.82279235618497936441431889487, −4.72569401628877931993790594251, −4.53621278437423131156441242397, −4.50446652111217597549946894911, −4.20479579491722006989347951499, −4.18257211885833806088839208388, −3.74227391064698335866346748139, −3.18451992999944285344997090915, −2.85580381149248382211547205684, −2.82443283507904852970974728313, −2.47011555471893639618725310927, −2.32257314621944695857693257046, −2.22489985473569677579526777026, −2.16674946656727430478186301230, −1.01860698723468507585142813022, −0.853426987254315009453191582461, −0.73113531555575922734967081888, −0.41749332664145819041727744440,
0.41749332664145819041727744440, 0.73113531555575922734967081888, 0.853426987254315009453191582461, 1.01860698723468507585142813022, 2.16674946656727430478186301230, 2.22489985473569677579526777026, 2.32257314621944695857693257046, 2.47011555471893639618725310927, 2.82443283507904852970974728313, 2.85580381149248382211547205684, 3.18451992999944285344997090915, 3.74227391064698335866346748139, 4.18257211885833806088839208388, 4.20479579491722006989347951499, 4.50446652111217597549946894911, 4.53621278437423131156441242397, 4.72569401628877931993790594251, 4.82279235618497936441431889487, 5.40182182218295441464243685702, 5.53899519571204408873603913648, 5.81582228857043371436225142245, 5.91083659788404491074606441399, 5.98890025400125078948367871776, 6.33524267824542268484464267993, 6.60811824940103760070520816662
