| L(s) = 1 | + 8·5-s − 2·9-s + 8·11-s + 8·13-s + 7·16-s − 8·17-s − 12·19-s + 32·25-s − 8·29-s − 12·31-s + 12·37-s − 16·41-s − 16·45-s + 16·47-s + 10·49-s − 8·53-s + 64·55-s − 8·59-s + 64·65-s + 4·67-s + 12·73-s + 8·79-s + 56·80-s + 3·81-s − 8·83-s − 64·85-s + 8·89-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 2/3·9-s + 2.41·11-s + 2.21·13-s + 7/4·16-s − 1.94·17-s − 2.75·19-s + 32/5·25-s − 1.48·29-s − 2.15·31-s + 1.97·37-s − 2.49·41-s − 2.38·45-s + 2.33·47-s + 10/7·49-s − 1.09·53-s + 8.62·55-s − 1.04·59-s + 7.93·65-s + 0.488·67-s + 1.40·73-s + 0.900·79-s + 6.26·80-s + 1/3·81-s − 0.878·83-s − 6.94·85-s + 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.478795941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.478795941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 56 T^{3} + 82 T^{4} - 56 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 1214 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 442 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 444 T^{3} + 2702 T^{4} + 444 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 84 T^{3} - 802 T^{4} - 84 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6678 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1072 T^{3} + 8578 T^{4} - 1072 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 440 T^{3} + 6034 T^{4} + 440 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 156 T^{3} - 8194 T^{4} + 156 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 6818 T^{4} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 2798 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 632 T^{3} + 12466 T^{4} + 632 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 804 T^{3} + 8078 T^{4} - 804 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.752204616803414073277977067793, −8.671268309925956163172455826375, −8.240001933471805795657655140962, −8.206372225224573619202379033956, −7.62031941751579119384226152799, −7.28689479760363063113275329645, −6.65106633912813387570766308701, −6.56945405687456100145596481715, −6.50603376781577016233424553217, −6.45526391812804286070420999543, −5.94083086041956919488307324277, −5.76977898245887487808557883787, −5.60823923314576919235415092196, −5.50422801341887080606689785023, −4.99964617995686551287702773918, −4.44096407113185634962311689770, −3.99929676968126252418787321044, −3.95692633474812991662861989877, −3.68541019113202767043097232560, −3.12835913266333210364713736769, −2.50355439749296610744771599728, −2.17541129203978597405439983548, −1.88392866495815513188344540944, −1.56181998106492146980760416093, −1.22317055223549272782744152085,
1.22317055223549272782744152085, 1.56181998106492146980760416093, 1.88392866495815513188344540944, 2.17541129203978597405439983548, 2.50355439749296610744771599728, 3.12835913266333210364713736769, 3.68541019113202767043097232560, 3.95692633474812991662861989877, 3.99929676968126252418787321044, 4.44096407113185634962311689770, 4.99964617995686551287702773918, 5.50422801341887080606689785023, 5.60823923314576919235415092196, 5.76977898245887487808557883787, 5.94083086041956919488307324277, 6.45526391812804286070420999543, 6.50603376781577016233424553217, 6.56945405687456100145596481715, 6.65106633912813387570766308701, 7.28689479760363063113275329645, 7.62031941751579119384226152799, 8.206372225224573619202379033956, 8.240001933471805795657655140962, 8.671268309925956163172455826375, 8.752204616803414073277977067793
