| L(s) = 1 | + 4·2-s + 4·4-s − 12·5-s − 16·8-s − 48·10-s + 4·11-s − 96·13-s − 64·16-s − 132·17-s + 120·19-s − 48·20-s + 16·22-s − 76·23-s + 284·25-s − 384·26-s + 224·29-s + 432·31-s − 64·32-s − 528·34-s + 280·37-s + 480·38-s + 192·40-s + 72·41-s − 256·43-s + 16·44-s − 304·46-s + 264·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 1.07·5-s − 0.707·8-s − 1.51·10-s + 0.109·11-s − 2.04·13-s − 16-s − 1.88·17-s + 1.44·19-s − 0.536·20-s + 0.155·22-s − 0.689·23-s + 2.27·25-s − 2.89·26-s + 1.43·29-s + 2.50·31-s − 0.353·32-s − 2.66·34-s + 1.24·37-s + 2.04·38-s + 0.758·40-s + 0.274·41-s − 0.907·43-s + 0.0548·44-s − 0.974·46-s + 0.819·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.575272960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.575272960\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 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
−6.95281073799891623302048060598, −6.52117307653332419221556341967, −6.30155404260583660587168673293, −6.28182422634292639031004839689, −5.88668546350270029451331823472, −5.80484605841486410191190142055, −5.09570395837229049823408273628, −5.06357129771516467069248785601, −4.99629848373123895943122942232, −4.63050279976944092648708099988, −4.52657336495654746081666719398, −4.50670973592099832736523013933, −3.99438367542033074603100765357, −3.93498955366273542526212118252, −3.38331683814579567936102602662, −3.17223837775400946580165782771, −3.05920866239649841382141776416, −2.60778978049516461628672233542, −2.57845861339206173677036291346, −2.08952014264613264108021015365, −2.00703522574291181422174451336, −1.04836784094789635516759232681, −0.991519659398431442247503368102, −0.51197566317782901296587770156, −0.36127376845765780076231228750,
0.36127376845765780076231228750, 0.51197566317782901296587770156, 0.991519659398431442247503368102, 1.04836784094789635516759232681, 2.00703522574291181422174451336, 2.08952014264613264108021015365, 2.57845861339206173677036291346, 2.60778978049516461628672233542, 3.05920866239649841382141776416, 3.17223837775400946580165782771, 3.38331683814579567936102602662, 3.93498955366273542526212118252, 3.99438367542033074603100765357, 4.50670973592099832736523013933, 4.52657336495654746081666719398, 4.63050279976944092648708099988, 4.99629848373123895943122942232, 5.06357129771516467069248785601, 5.09570395837229049823408273628, 5.80484605841486410191190142055, 5.88668546350270029451331823472, 6.28182422634292639031004839689, 6.30155404260583660587168673293, 6.52117307653332419221556341967, 6.95281073799891623302048060598
