| L(s) = 1 | + 3·2-s − 94·3-s + 42·4-s − 330·5-s − 282·6-s − 417·8-s + 5.71e3·9-s − 990·10-s − 2.84e3·11-s − 3.94e3·12-s + 5.06e3·13-s + 3.10e4·15-s + 1.39e4·16-s + 1.48e3·17-s + 1.71e4·18-s − 3.28e4·19-s − 1.38e4·20-s − 8.53e3·22-s + 6.57e3·23-s + 3.91e4·24-s + 1.61e5·25-s + 1.52e4·26-s − 3.70e5·27-s + 4.12e4·29-s + 9.30e4·30-s + 3.91e5·31-s − 3.36e4·32-s + ⋯ |
| L(s) = 1 | + 0.265·2-s − 2.01·3-s + 0.328·4-s − 1.18·5-s − 0.532·6-s − 0.287·8-s + 2.61·9-s − 0.313·10-s − 0.644·11-s − 0.659·12-s + 0.639·13-s + 2.37·15-s + 0.853·16-s + 0.0734·17-s + 0.693·18-s − 1.09·19-s − 0.387·20-s − 0.170·22-s + 0.112·23-s + 0.578·24-s + 2.07·25-s + 0.169·26-s − 3.62·27-s + 0.314·29-s + 0.629·30-s + 2.36·31-s − 0.181·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5764801 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5764801 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.009194601971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.009194601971\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 T - 33 T^{2} + 321 p T^{3} - 3943 p^{2} T^{4} + 321 p^{8} T^{5} - 33 p^{14} T^{6} - 3 p^{21} T^{7} + p^{28} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 94 T + 3118 T^{2} + 42112 p T^{3} + 954247 p^{2} T^{4} + 42112 p^{8} T^{5} + 3118 p^{14} T^{6} + 94 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 66 p T - 2118 p^{2} T^{2} + 14784 p^{3} T^{3} + 18534551 p^{4} T^{4} + 14784 p^{10} T^{5} - 2118 p^{16} T^{6} + 66 p^{22} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2844 T - 29998230 T^{2} - 2524834944 T^{3} + 913217889351659 T^{4} - 2524834944 p^{7} T^{5} - 29998230 p^{14} T^{6} + 2844 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 2534 T - 41123742 T^{2} - 2534 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 1488 T - 796315678 T^{2} + 32955515712 T^{3} + 468363501153159843 T^{4} + 32955515712 p^{7} T^{5} - 796315678 p^{14} T^{6} - 1488 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 32810 T - 821175938 T^{2} + 3606756053600 T^{3} + 1839903037090471423 T^{4} + 3606756053600 p^{7} T^{5} - 821175938 p^{14} T^{6} + 32810 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6576 T - 6776719822 T^{2} - 67816341504 T^{3} + 34771829644557187923 T^{4} - 67816341504 p^{7} T^{5} - 6776719822 p^{14} T^{6} - 6576 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 20640 T + 15579628518 T^{2} - 20640 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 391836 T + 61486586290 T^{2} - 14507193586161024 T^{3} + \)\(34\!\cdots\!59\)\( T^{4} - 14507193586161024 p^{7} T^{5} + 61486586290 p^{14} T^{6} - 391836 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 367392 T + 71124113482 T^{2} - 46295428199372928 T^{3} - \)\(17\!\cdots\!77\)\( T^{4} - 46295428199372928 p^{7} T^{5} + 71124113482 p^{14} T^{6} + 367392 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 734664 T + 402811824126 T^{2} - 734664 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 480476 T + 594501933318 T^{2} + 480476 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1089108 T + 170819098322 T^{2} - 2277212013836928 T^{3} + \)\(17\!\cdots\!63\)\( T^{4} - 2277212013836928 p^{7} T^{5} + 170819098322 p^{14} T^{6} - 1089108 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 2858844 T + 3786130953938 T^{2} + 5824711062825811056 T^{3} + \)\(82\!\cdots\!03\)\( T^{4} + 5824711062825811056 p^{7} T^{5} + 3786130953938 p^{14} T^{6} + 2858844 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 160170 T - 4336274439298 T^{2} - 98564469827644800 T^{3} + \)\(12\!\cdots\!43\)\( T^{4} - 98564469827644800 p^{7} T^{5} - 4336274439298 p^{14} T^{6} + 160170 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 864646 T - 5008356465590 T^{2} + 457844324641237856 T^{3} + \)\(20\!\cdots\!59\)\( T^{4} + 457844324641237856 p^{7} T^{5} - 5008356465590 p^{14} T^{6} - 864646 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 328648 T - 11479536848678 T^{2} + 175457560334425472 T^{3} + \)\(96\!\cdots\!03\)\( T^{4} + 175457560334425472 p^{7} T^{5} - 11479536848678 p^{14} T^{6} - 328648 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 7500216 T + 28549732695406 T^{2} + 7500216 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4301244 T - 253409836502 T^{2} - 14369111007814970064 T^{3} + \)\(25\!\cdots\!43\)\( T^{4} - 14369111007814970064 p^{7} T^{5} - 253409836502 p^{14} T^{6} + 4301244 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6408440 T + 8995491286882 T^{2} + 40598787702696064000 T^{3} - \)\(12\!\cdots\!57\)\( T^{4} + 40598787702696064000 p^{7} T^{5} + 8995491286882 p^{14} T^{6} - 6408440 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 11659074 T + 84453675852838 T^{2} - 11659074 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 9772260 T + 11684070451002 T^{2} - 45437830420574079600 T^{3} + \)\(84\!\cdots\!63\)\( T^{4} - 45437830420574079600 p^{7} T^{5} + 11684070451002 p^{14} T^{6} + 9772260 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 10762752 T + 188617737573662 T^{2} - 10762752 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−10.32414483489172003628331996400, −10.18944660592757677094838528661, −9.427429547037717235024023194464, −8.997978190536007208183678678370, −8.991030869677108127569912963837, −8.142213158378598940137063543049, −8.102266347097256257706145081526, −7.64216893711663855605284134699, −7.46246722991508891355055170505, −6.75497765690762189786919674211, −6.70367444730053676539023240613, −6.22063181565621774895178007957, −5.88192375036138314900487261931, −5.76549339113643247114425151039, −5.00927588556670901337779833130, −4.84825714580521549277265772678, −4.41326621161018922925570076707, −4.09412423636808436741260114283, −3.50765660173953933289187791823, −3.06266857769061002064685138622, −2.55567370112560081484936489261, −1.75726161055369082433443254210, −0.982532596214772184443320792706, −0.928293904037008866802961245610, −0.02508879449854715529057705770,
0.02508879449854715529057705770, 0.928293904037008866802961245610, 0.982532596214772184443320792706, 1.75726161055369082433443254210, 2.55567370112560081484936489261, 3.06266857769061002064685138622, 3.50765660173953933289187791823, 4.09412423636808436741260114283, 4.41326621161018922925570076707, 4.84825714580521549277265772678, 5.00927588556670901337779833130, 5.76549339113643247114425151039, 5.88192375036138314900487261931, 6.22063181565621774895178007957, 6.70367444730053676539023240613, 6.75497765690762189786919674211, 7.46246722991508891355055170505, 7.64216893711663855605284134699, 8.102266347097256257706145081526, 8.142213158378598940137063543049, 8.991030869677108127569912963837, 8.997978190536007208183678678370, 9.427429547037717235024023194464, 10.18944660592757677094838528661, 10.32414483489172003628331996400
