| L(s) = 1 | + 4·2-s + 4·4-s + 5·5-s − 16·8-s + 20·10-s + 67·11-s + 82·13-s − 64·16-s − 92·17-s − 43·19-s + 20·20-s + 268·22-s + 148·23-s − 80·25-s + 328·26-s − 154·29-s + 520·31-s − 64·32-s − 368·34-s − 7·37-s − 172·38-s − 80·40-s + 852·41-s − 214·43-s + 268·44-s + 592·46-s + 576·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.447·5-s − 0.707·8-s + 0.632·10-s + 1.83·11-s + 1.74·13-s − 16-s − 1.31·17-s − 0.519·19-s + 0.223·20-s + 2.59·22-s + 1.34·23-s − 0.639·25-s + 2.47·26-s − 0.986·29-s + 3.01·31-s − 0.353·32-s − 1.85·34-s − 0.0311·37-s − 0.734·38-s − 0.316·40-s + 3.24·41-s − 0.758·43-s + 0.918·44-s + 1.89·46-s + 1.78·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.43922904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.43922904\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 659 T^{2} + p^{6} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - p T + 21 p T^{2} + 66 p^{2} T^{3} - 494 p^{2} T^{4} + 66 p^{5} T^{5} + 21 p^{7} T^{6} - p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 67 T + 1041 T^{2} - 52662 T^{3} + 4142284 T^{4} - 52662 p^{3} T^{5} + 1041 p^{6} T^{6} - 67 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 41 T + 4478 T^{2} - 41 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 92 T + 1902 T^{2} - 17664 p T^{3} - 78413 p^{2} T^{4} - 17664 p^{4} T^{5} + 1902 p^{6} T^{6} + 92 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 43 T - 9305 T^{2} - 110252 T^{3} + 64683544 T^{4} - 110252 p^{3} T^{5} - 9305 p^{6} T^{6} + 43 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 148 T - 2526 T^{2} - 14208 T^{3} + 182283043 T^{4} - 14208 p^{3} T^{5} - 2526 p^{6} T^{6} - 148 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 77 T + 9574 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 520 T + 144563 T^{2} - 34452600 T^{3} + 6891960488 T^{4} - 34452600 p^{3} T^{5} + 144563 p^{6} T^{6} - 520 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 7 T - 74033 T^{2} - 190568 T^{3} + 2919934318 T^{4} - 190568 p^{3} T^{5} - 74033 p^{6} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 426 T + 171106 T^{2} - 426 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 107 T + 86220 T^{2} + 107 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 576 T + 89606 T^{2} - 19885824 T^{3} + 13421113923 T^{4} - 19885824 p^{3} T^{5} + 89606 p^{6} T^{6} - 576 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 243 T - 250441 T^{2} + 2851848 T^{3} + 64828660998 T^{4} + 2851848 p^{3} T^{5} - 250441 p^{6} T^{6} + 243 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 7 T - 200565 T^{2} - 1471008 T^{3} - 1944620216 T^{4} - 1471008 p^{3} T^{5} - 200565 p^{6} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 224 T - 410950 T^{2} + 1604736 T^{3} + 149727814859 T^{4} + 1604736 p^{3} T^{5} - 410950 p^{6} T^{6} + 224 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 687 T - 206863 T^{2} + 53109222 T^{3} + 228403689708 T^{4} + 53109222 p^{3} T^{5} - 206863 p^{6} T^{6} + 687 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 472 T + 637018 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 921 T + 270053 T^{2} + 184058166 T^{3} - 147013032042 T^{4} + 184058166 p^{3} T^{5} + 270053 p^{6} T^{6} - 921 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 526 T - 757051 T^{2} - 25063374 T^{3} + 689091996644 T^{4} - 25063374 p^{3} T^{5} - 757051 p^{6} T^{6} - 526 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 221 T + 945628 T^{2} - 221 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 774 T - 367486 T^{2} + 343173024 T^{3} + 14930800239 T^{4} + 343173024 p^{3} T^{5} - 367486 p^{6} T^{6} - 774 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 1953 T + 2366992 T^{2} - 1953 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−9.157778419961624937348124009170, −9.156192884005659257655651105353, −8.758267584146240670953131384352, −8.749474606451993875632562555088, −8.458443446236150373229732401497, −7.57156404509733251522889581131, −7.56260190898312750518822716778, −7.49449893909172609093459523426, −6.70577882679436627329826763631, −6.34382316348076456163316352552, −6.30217464708227521099741933644, −6.10982176661184458065006393916, −5.95061352504484639123342760477, −5.41714158738784248193475557107, −4.80728460731211950432347962366, −4.61796203369642929044808974706, −4.47008788633277261006800360916, −3.85554518474414186548152917028, −3.75092871780324631700017926096, −3.49780124844820060528400310886, −2.59431671091595671232273793524, −2.54644393266645707819597182219, −1.82365742744416616259945670866, −0.958581059815594624918747057432, −0.837399872145892182213605439823,
0.837399872145892182213605439823, 0.958581059815594624918747057432, 1.82365742744416616259945670866, 2.54644393266645707819597182219, 2.59431671091595671232273793524, 3.49780124844820060528400310886, 3.75092871780324631700017926096, 3.85554518474414186548152917028, 4.47008788633277261006800360916, 4.61796203369642929044808974706, 4.80728460731211950432347962366, 5.41714158738784248193475557107, 5.95061352504484639123342760477, 6.10982176661184458065006393916, 6.30217464708227521099741933644, 6.34382316348076456163316352552, 6.70577882679436627329826763631, 7.49449893909172609093459523426, 7.56260190898312750518822716778, 7.57156404509733251522889581131, 8.458443446236150373229732401497, 8.749474606451993875632562555088, 8.758267584146240670953131384352, 9.156192884005659257655651105353, 9.157778419961624937348124009170
