| L(s) = 1 | − 6·2-s − 11·3-s + 5·4-s − 29·5-s + 66·6-s − 32·7-s + 28·8-s + 7·9-s + 174·10-s − 11·11-s − 55·12-s − 45·13-s + 192·14-s + 319·15-s + 71·16-s − 56·17-s − 42·18-s − 144·19-s − 145·20-s + 352·21-s + 66·22-s − 275·23-s − 308·24-s + 216·25-s + 270·26-s + 375·27-s − 160·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.11·3-s + 5/8·4-s − 2.59·5-s + 4.49·6-s − 1.72·7-s + 1.23·8-s + 7/27·9-s + 5.50·10-s − 0.301·11-s − 1.32·12-s − 0.960·13-s + 3.66·14-s + 5.49·15-s + 1.10·16-s − 0.798·17-s − 0.549·18-s − 1.73·19-s − 1.62·20-s + 3.65·21-s + 0.639·22-s − 2.49·23-s − 2.61·24-s + 1.72·25-s + 2.03·26-s + 2.67·27-s − 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1874161 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1874161 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 37 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 3 p T + 31 T^{2} + p^{7} T^{3} + 187 p T^{4} + p^{10} T^{5} + 31 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 11 T + 38 p T^{2} + 802 T^{3} + 4721 T^{4} + 802 p^{3} T^{5} + 38 p^{7} T^{6} + 11 p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 29 T + p^{4} T^{2} + 9203 T^{3} + 116408 T^{4} + 9203 p^{3} T^{5} + p^{10} T^{6} + 29 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 32 T + 405 T^{2} - 6690 T^{3} - 222792 T^{4} - 6690 p^{3} T^{5} + 405 p^{6} T^{6} + 32 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + p T + 3892 T^{2} + 25622 T^{3} + 6986699 T^{4} + 25622 p^{3} T^{5} + 3892 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 45 T + 4153 T^{2} - 11005 T^{3} + 3964072 T^{4} - 11005 p^{3} T^{5} + 4153 p^{6} T^{6} + 45 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 56 T + 19996 T^{2} + 817160 T^{3} + 148203782 T^{4} + 817160 p^{3} T^{5} + 19996 p^{6} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 144 T + 13796 T^{2} + 1744 T^{3} - 20218442 T^{4} + 1744 p^{3} T^{5} + 13796 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 275 T + 61129 T^{2} + 8508911 T^{3} + 1097840624 T^{4} + 8508911 p^{3} T^{5} + 61129 p^{6} T^{6} + 275 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + p T + 73041 T^{2} + 358675 T^{3} + 2352225388 T^{4} + 358675 p^{3} T^{5} + 73041 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 111 T + 78499 T^{2} - 7493995 T^{3} + 3345698920 T^{4} - 7493995 p^{3} T^{5} + 78499 p^{6} T^{6} - 111 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 9 T + 219722 T^{2} - 701352 T^{3} + 21026186889 T^{4} - 701352 p^{3} T^{5} + 219722 p^{6} T^{6} + 9 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 190 T + 276072 T^{2} + 43760974 T^{3} + 31374739038 T^{4} + 43760974 p^{3} T^{5} + 276072 p^{6} T^{6} + 190 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 472 T + 274781 T^{2} + 83633366 T^{3} + 30614291256 T^{4} + 83633366 p^{3} T^{5} + 274781 p^{6} T^{6} + 472 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 6 p T + 590617 T^{2} - 2677270 p T^{3} + 131531367356 T^{4} - 2677270 p^{4} T^{5} + 590617 p^{6} T^{6} - 6 p^{10} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 1530 T + 1509296 T^{2} + 988232778 T^{3} + 515754398670 T^{4} + 988232778 p^{3} T^{5} + 1509296 p^{6} T^{6} + 1530 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 31 T + 503293 T^{2} + 97400849 T^{3} + 123460181648 T^{4} + 97400849 p^{3} T^{5} + 503293 p^{6} T^{6} + 31 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 5 T + 969017 T^{2} - 27202963 T^{3} + 405557036068 T^{4} - 27202963 p^{3} T^{5} + 969017 p^{6} T^{6} + 5 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 390 T + 883201 T^{2} - 193050418 T^{3} + 354918917648 T^{4} - 193050418 p^{3} T^{5} + 883201 p^{6} T^{6} - 390 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 967 T + 1414746 T^{2} + 888044448 T^{3} + 787959172037 T^{4} + 888044448 p^{3} T^{5} + 1414746 p^{6} T^{6} + 967 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 17 T + 1318969 T^{2} + 241580635 T^{3} + 785040686656 T^{4} + 241580635 p^{3} T^{5} + 1318969 p^{6} T^{6} - 17 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 3138 T + 5313901 T^{2} + 6108524926 T^{3} + 5276627905340 T^{4} + 6108524926 p^{3} T^{5} + 5313901 p^{6} T^{6} + 3138 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 224 T + 1130120 T^{2} + 8586928 p T^{3} + 386569796910 T^{4} + 8586928 p^{4} T^{5} + 1130120 p^{6} T^{6} - 224 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2532 T + 5647288 T^{2} + 7419989228 T^{3} + 8645712561262 T^{4} + 7419989228 p^{3} T^{5} + 5647288 p^{6} T^{6} + 2532 p^{9} T^{7} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.23135140140682132377372743030, −12.05840553486732605241689984041, −11.60468000127913218752611070990, −11.42351641201810620196718023965, −11.33119953615576540883418375373, −10.80840236921413907574507322412, −10.23873620884608393872252679169, −10.05818344811448326782624128105, −10.00832145388813994496691646004, −9.462703982478116740296473652350, −8.941858707706378091678447767642, −8.780468523866910250778532040331, −8.493660604182453818254181387869, −7.900875555567785917514147908766, −7.78376681508433605650012954279, −7.75526859384778935662425146475, −6.65037594061025140250680699744, −6.52302048145306040180669358089, −5.87351061841605176309591106106, −5.86282248255721465636512227275, −5.22168365765523004804521381858, −4.26946913569808664984842704245, −4.24568600511090246463778800004, −3.54833599290655723808860104253, −2.81990088922273113453719976885, 0, 0, 0, 0,
2.81990088922273113453719976885, 3.54833599290655723808860104253, 4.24568600511090246463778800004, 4.26946913569808664984842704245, 5.22168365765523004804521381858, 5.86282248255721465636512227275, 5.87351061841605176309591106106, 6.52302048145306040180669358089, 6.65037594061025140250680699744, 7.75526859384778935662425146475, 7.78376681508433605650012954279, 7.900875555567785917514147908766, 8.493660604182453818254181387869, 8.780468523866910250778532040331, 8.941858707706378091678447767642, 9.462703982478116740296473652350, 10.00832145388813994496691646004, 10.05818344811448326782624128105, 10.23873620884608393872252679169, 10.80840236921413907574507322412, 11.33119953615576540883418375373, 11.42351641201810620196718023965, 11.60468000127913218752611070990, 12.05840553486732605241689984041, 12.23135140140682132377372743030
