| L(s) = 1 | − 20·5-s − 6·9-s + 32·13-s + 20·17-s + 192·25-s − 56·29-s − 208·37-s + 92·41-s + 120·45-s − 14·49-s + 112·53-s + 168·61-s − 640·65-s − 32·73-s + 27·81-s − 400·85-s + 236·89-s − 304·97-s + 164·101-s + 488·109-s + 136·113-s − 192·117-s + 80·121-s − 1.26e3·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4·5-s − 2/3·9-s + 2.46·13-s + 1.17·17-s + 7.67·25-s − 1.93·29-s − 5.62·37-s + 2.24·41-s + 8/3·45-s − 2/7·49-s + 2.11·53-s + 2.75·61-s − 9.84·65-s − 0.438·73-s + 1/3·81-s − 4.70·85-s + 2.65·89-s − 3.13·97-s + 1.62·101-s + 4.47·109-s + 1.20·113-s − 1.64·117-s + 0.661·121-s − 10.0·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9721694137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9721694137\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 + 2 p T + 54 T^{2} + 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 11982 T^{4} - 80 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 16 T + 318 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 10 T + 582 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 65742 T^{4} - 128 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 28 T + 1542 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1490 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 104 T + 5358 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 46 T + 2862 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6116 T^{2} + 15844902 T^{4} - 6116 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8308 T^{2} + 27002982 T^{4} - 8308 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 56 T + 2286 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1940 T^{2} + 10358022 T^{4} - 1940 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 84 T + 7862 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 44 T^{2} + 31484742 T^{4} + 44 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 13280 T^{2} + 92626062 T^{4} - 13280 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 8622 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 17524 T^{2} + 151177062 T^{4} - 17524 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 13124 T^{2} + 100043430 T^{4} - 13124 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 118 T + 18798 T^{2} - 118 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 152 T + 22494 T^{2} + 152 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.277230148425120898056742559265, −7.78757809077339233482127021946, −7.72398863018827001195883089454, −7.53387843968949352616248578625, −7.14201773731197661869888113690, −7.02037292746579154993273889394, −6.97764077593956896849042294250, −6.33750489192994021386882951899, −6.15662948415798199815158863989, −5.63757409721672975319148220829, −5.60730307768231683222680155054, −5.24001782410416655106249149361, −4.95959737520812716083064490399, −4.53854952614387676529738371593, −4.07915350455628001470979664308, −3.90787895933639662129369703757, −3.62291412141551214660322493486, −3.57544154818294840941504664044, −3.46072622982116272887091271230, −3.12171898320677064943659043819, −2.37394789608776277719178775859, −1.85721004119262984279634473062, −1.27947823532545643334349461759, −0.54052472070413698371622727401, −0.45265379317804762397167901961,
0.45265379317804762397167901961, 0.54052472070413698371622727401, 1.27947823532545643334349461759, 1.85721004119262984279634473062, 2.37394789608776277719178775859, 3.12171898320677064943659043819, 3.46072622982116272887091271230, 3.57544154818294840941504664044, 3.62291412141551214660322493486, 3.90787895933639662129369703757, 4.07915350455628001470979664308, 4.53854952614387676529738371593, 4.95959737520812716083064490399, 5.24001782410416655106249149361, 5.60730307768231683222680155054, 5.63757409721672975319148220829, 6.15662948415798199815158863989, 6.33750489192994021386882951899, 6.97764077593956896849042294250, 7.02037292746579154993273889394, 7.14201773731197661869888113690, 7.53387843968949352616248578625, 7.72398863018827001195883089454, 7.78757809077339233482127021946, 8.277230148425120898056742559265
