| L(s) = 1 | + 8·5-s + 16·16-s − 24·23-s − 36·25-s + 112·31-s + 80·37-s + 8·47-s + 192·49-s + 40·53-s + 64·59-s − 160·67-s + 200·71-s + 128·80-s − 144·89-s − 140·97-s − 76·103-s + 392·113-s − 192·115-s − 552·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 896·155-s + 157-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 16-s − 1.04·23-s − 1.43·25-s + 3.61·31-s + 2.16·37-s + 8/47·47-s + 3.91·49-s + 0.754·53-s + 1.08·59-s − 2.38·67-s + 2.81·71-s + 8/5·80-s − 1.61·89-s − 1.44·97-s − 0.737·103-s + 3.46·113-s − 1.66·115-s − 4.41·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 5.78·155-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.835869990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.835869990\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 4 T + 42 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 14015 T^{4} - 192 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 16 p T^{2} + 52386 T^{4} - 16 p^{5} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 820 T^{2} + 307494 T^{4} - 820 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1120 T^{2} + 554559 T^{4} - 1120 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 12 T - 106 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 108 T^{2} + 1403606 T^{4} + 108 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 56 T + 2703 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 40 T + 615 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6100 T^{2} + 14856822 T^{4} - 6100 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5008 T^{2} + 13022946 T^{4} - 5008 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T - 378 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 20 T + 2646 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 32 T + 6918 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6528 T^{2} + 37724303 T^{4} - 6528 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 80 T + 8991 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 100 T + 11994 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12720 T^{2} + 78809759 T^{4} - 12720 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16816 T^{2} + 134402751 T^{4} - 16816 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1908 T^{2} - 14635114 T^{4} - 1908 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 72 T + 16706 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 70 T + 9243 T^{2} + 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.81038833563470496632046855280, −6.45660493008705657432841958935, −6.38624850670270311294707009368, −6.10300704335667886734539125592, −5.93989516502420001503716857814, −5.72375915606075983748954179142, −5.58656197331143256994246627962, −5.51996743432892389448272729943, −5.15950501073894553218895071989, −4.78068391169397937523921031814, −4.38035061555196622859593323936, −4.32911882600920614929997647698, −4.14591216911879078774303876576, −3.82789021532231255821015445133, −3.69370510571074994563958899804, −3.07440597806449327624090637901, −2.80627759646076993148797871491, −2.77261047221953644981915869415, −2.24133162149808805664573584350, −2.21738633539039472129397783641, −1.89347065269564706961218406159, −1.47165042827806192874556241709, −0.948941913927588368692577899269, −0.837949510036322881912168919834, −0.45919069746965776049783408575,
0.45919069746965776049783408575, 0.837949510036322881912168919834, 0.948941913927588368692577899269, 1.47165042827806192874556241709, 1.89347065269564706961218406159, 2.21738633539039472129397783641, 2.24133162149808805664573584350, 2.77261047221953644981915869415, 2.80627759646076993148797871491, 3.07440597806449327624090637901, 3.69370510571074994563958899804, 3.82789021532231255821015445133, 4.14591216911879078774303876576, 4.32911882600920614929997647698, 4.38035061555196622859593323936, 4.78068391169397937523921031814, 5.15950501073894553218895071989, 5.51996743432892389448272729943, 5.58656197331143256994246627962, 5.72375915606075983748954179142, 5.93989516502420001503716857814, 6.10300704335667886734539125592, 6.38624850670270311294707009368, 6.45660493008705657432841958935, 6.81038833563470496632046855280
