| L(s) = 1 | + 2·4-s − 16·7-s + 24·13-s + 11·16-s − 32·28-s − 56·31-s − 152·37-s + 48·43-s − 16·49-s + 48·52-s − 168·61-s + 68·64-s − 208·67-s − 216·73-s − 56·79-s − 384·91-s + 232·97-s + 400·103-s − 280·109-s − 176·112-s + 248·121-s − 112·124-s + 127-s + 131-s + 137-s + 139-s − 304·148-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2.28·7-s + 1.84·13-s + 0.687·16-s − 8/7·28-s − 1.80·31-s − 4.10·37-s + 1.11·43-s − 0.326·49-s + 0.923·52-s − 2.75·61-s + 1.06·64-s − 3.10·67-s − 2.95·73-s − 0.708·79-s − 4.21·91-s + 2.39·97-s + 3.88·103-s − 2.56·109-s − 1.57·112-s + 2.04·121-s − 0.903·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2.05·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0009728820865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0009728820865\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} - 7 T^{4} - p^{5} T^{6} + p^{8} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 8 T + 104 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 248 T^{2} + 36818 T^{4} - 248 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 12 T + 124 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 452 T^{2} + 177158 T^{4} - 452 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 682 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 p T^{2} + 146022 T^{4} - 20 p^{5} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1340 T^{2} + 839462 T^{4} - 1340 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 28 T + 158 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 76 T + 3372 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6080 T^{2} + 14890562 T^{4} - 6080 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 24 T + 2842 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6740 T^{2} + 20806502 T^{4} - 6740 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8660 T^{2} + 33290822 T^{4} - 8660 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7928 T^{2} + 34352978 T^{4} - 7928 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 84 T + 9166 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 104 T + 11042 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 740 T^{2} + 49721222 T^{4} - 740 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 108 T + 13534 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 28 T + 5918 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20500 T^{2} + 192514182 T^{4} - 20500 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 17248 T^{2} + 181143618 T^{4} - 17248 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 116 T + 18942 T^{2} - 116 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.771540508456542257439874828543, −8.763283673038257514687933381052, −8.098363066238493328396261624517, −7.967456245326947934849873386793, −7.49712896268192577298206420800, −7.27232540822787700322405689974, −7.07656889872283342617503770500, −6.90105336670096356492159986364, −6.40895329536573999234444378921, −6.18721668788444435727139468612, −6.07223660799818474551369382295, −5.82788408101423982083400941443, −5.60387729872354185378009938888, −4.96757896607174629611498454778, −4.90897585004143669745059241703, −4.21594390502300511314199825844, −4.01840945185131103977941563812, −3.39183471194962407818964197519, −3.33419183798934732100153310459, −3.20340038075304916193212809775, −2.92884811075816153625341722371, −1.92048620185143657733871544169, −1.74875175565402326788787094725, −1.24151582840052664049767444817, −0.008502606129613530832283298733,
0.008502606129613530832283298733, 1.24151582840052664049767444817, 1.74875175565402326788787094725, 1.92048620185143657733871544169, 2.92884811075816153625341722371, 3.20340038075304916193212809775, 3.33419183798934732100153310459, 3.39183471194962407818964197519, 4.01840945185131103977941563812, 4.21594390502300511314199825844, 4.90897585004143669745059241703, 4.96757896607174629611498454778, 5.60387729872354185378009938888, 5.82788408101423982083400941443, 6.07223660799818474551369382295, 6.18721668788444435727139468612, 6.40895329536573999234444378921, 6.90105336670096356492159986364, 7.07656889872283342617503770500, 7.27232540822787700322405689974, 7.49712896268192577298206420800, 7.967456245326947934849873386793, 8.098363066238493328396261624517, 8.763283673038257514687933381052, 8.771540508456542257439874828543
