| L(s) = 1 | − 4·2-s + 6·3-s + 4·4-s + 12·5-s − 24·6-s + 16·8-s + 9·9-s − 48·10-s − 4·11-s + 24·12-s − 96·13-s + 72·15-s − 64·16-s + 132·17-s − 36·18-s + 120·19-s + 48·20-s + 16·22-s + 76·23-s + 96·24-s + 284·25-s + 384·26-s − 54·27-s − 224·29-s − 288·30-s + 432·31-s + 64·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.07·5-s − 1.63·6-s + 0.707·8-s + 1/3·9-s − 1.51·10-s − 0.109·11-s + 0.577·12-s − 2.04·13-s + 1.23·15-s − 16-s + 1.88·17-s − 0.471·18-s + 1.44·19-s + 0.536·20-s + 0.155·22-s + 0.689·23-s + 0.816·24-s + 2.27·25-s + 2.89·26-s − 0.384·27-s − 1.43·29-s − 1.75·30-s + 2.50·31-s + 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.292171143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.292171143\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T - 28 p T^{2} - 408 T^{3} + 47031 T^{4} - 408 p^{3} T^{5} - 28 p^{7} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 2578 T^{2} - 272 T^{3} + 4935979 T^{4} - 272 p^{3} T^{5} - 2578 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 48 T + 4520 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 132 T + 3484 T^{2} - 31944 p T^{3} + 292671 p^{2} T^{4} - 31944 p^{4} T^{5} + 3484 p^{6} T^{6} - 132 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 120 T + 1314 T^{2} + 75840 T^{3} + 25427915 T^{4} + 75840 p^{3} T^{5} + 1314 p^{6} T^{6} - 120 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 76 T + 806 T^{2} + 1471664 T^{3} - 193611581 T^{4} + 1471664 p^{3} T^{5} + 806 p^{6} T^{6} - 76 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 112 T + 6914 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 432 T + 86218 T^{2} - 17635968 T^{3} + 3634145571 T^{4} - 17635968 p^{3} T^{5} + 86218 p^{6} T^{6} - 432 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 280 T + 22294 T^{2} + 12656000 T^{3} - 3389038373 T^{4} + 12656000 p^{3} T^{5} + 22294 p^{6} T^{6} - 280 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 36 T + 105908 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 128 T - 2778 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 264 T + 73114 T^{2} - 55720896 T^{3} - 18003580413 T^{4} - 55720896 p^{3} T^{5} + 73114 p^{6} T^{6} + 264 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 268 T - 170158 T^{2} - 14946896 T^{3} + 25697985547 T^{4} - 14946896 p^{3} T^{5} - 170158 p^{6} T^{6} + 268 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 336 T - 148478 T^{2} + 50193024 T^{3} + 2949366651 T^{4} + 50193024 p^{3} T^{5} - 148478 p^{6} T^{6} - 336 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 504 T - 14232 T^{2} - 93599856 T^{3} - 37220190553 T^{4} - 93599856 p^{3} T^{5} - 14232 p^{6} T^{6} + 504 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 384 T - 449462 T^{2} + 1769472 T^{3} + 221503407627 T^{4} + 1769472 p^{3} T^{5} - 449462 p^{6} T^{6} - 384 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 396 T + 521098 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 312 T + 3024 T^{2} - 213318768 T^{3} - 180306434737 T^{4} - 213318768 p^{3} T^{5} + 3024 p^{6} T^{6} + 312 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 848 T - 266750 T^{2} + 189952 T^{3} + 374274336739 T^{4} + 189952 p^{3} T^{5} - 266750 p^{6} T^{6} - 848 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 648 T + 1235750 T^{2} - 648 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 612 T - 1067780 T^{2} + 19820232 T^{3} + 1319275320879 T^{4} + 19820232 p^{3} T^{5} - 1067780 p^{6} T^{6} + 612 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 2184 T + 2982432 T^{2} - 2184 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.997495988082219928052138995501, −7.993582824165583565434479991128, −7.67024916278719500666449591141, −7.54211901128197272610444229434, −7.26062423748526715143701032325, −7.03737061123771432203602826896, −6.53688901837956724891541769642, −6.50180842657187499744186473756, −6.04140729435397959902325335882, −5.70140300065398609552938444192, −5.31910102700749610128031135464, −5.18854720306154847801049812726, −4.85616741808499269790988833193, −4.56693496811273228473113044933, −4.48958461364012067884978525250, −3.52558962983131916515414114309, −3.50045734769542530032275327851, −3.06886470268954972133931498690, −2.82009392903557096189334896203, −2.51503878560418890592643173052, −2.10410351785562661401817554255, −1.67056790923777975215390743256, −1.14342501457630612979047188323, −0.912251677926634142598715966588, −0.40979553908306752592611199821,
0.40979553908306752592611199821, 0.912251677926634142598715966588, 1.14342501457630612979047188323, 1.67056790923777975215390743256, 2.10410351785562661401817554255, 2.51503878560418890592643173052, 2.82009392903557096189334896203, 3.06886470268954972133931498690, 3.50045734769542530032275327851, 3.52558962983131916515414114309, 4.48958461364012067884978525250, 4.56693496811273228473113044933, 4.85616741808499269790988833193, 5.18854720306154847801049812726, 5.31910102700749610128031135464, 5.70140300065398609552938444192, 6.04140729435397959902325335882, 6.50180842657187499744186473756, 6.53688901837956724891541769642, 7.03737061123771432203602826896, 7.26062423748526715143701032325, 7.54211901128197272610444229434, 7.67024916278719500666449591141, 7.993582824165583565434479991128, 7.997495988082219928052138995501
