| L(s) = 1 | − 4·2-s + 8·4-s + 12·7-s − 8·8-s + 16·9-s + 4·11-s + 52·13-s − 48·14-s − 4·16-s + 12·17-s − 64·18-s − 24·19-s − 16·22-s + 104·23-s − 208·26-s + 96·28-s − 76·29-s + 60·31-s + 32·32-s − 48·34-s + 128·36-s − 68·37-s + 96·38-s + 64·41-s + 152·43-s + 32·44-s − 416·46-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·4-s + 12/7·7-s − 8-s + 16/9·9-s + 4/11·11-s + 4·13-s − 3.42·14-s − 1/4·16-s + 0.705·17-s − 3.55·18-s − 1.26·19-s − 0.727·22-s + 4.52·23-s − 8·26-s + 24/7·28-s − 2.62·29-s + 1.93·31-s + 32-s − 1.41·34-s + 32/9·36-s − 1.83·37-s + 2.52·38-s + 1.56·41-s + 3.53·43-s + 8/11·44-s − 9.04·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.845143163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.845143163\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 130 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 480 T^{3} + 3119 T^{4} - 480 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 480 T^{3} - 29281 T^{4} + 480 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 371 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 52 T + 1680 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 38 T + 1179 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 60 T + 1800 T^{2} - 83040 T^{3} + 3651983 T^{4} - 83040 p^{2} T^{5} + 1800 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 68 T + 2312 T^{2} + 103020 T^{3} + 4569134 T^{4} + 103020 p^{2} T^{5} + 2312 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 64 T + 2048 T^{2} - 29760 T^{3} - 1046206 T^{4} - 29760 p^{2} T^{5} + 2048 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 76 T + 4656 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} - 66912 T^{3} + 1577327 T^{4} - 66912 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2354 T^{2} + 13515891 T^{4} - 2354 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 52 T + 1352 T^{2} + 28704 T^{3} - 7969633 T^{4} + 28704 p^{2} T^{5} + 1352 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 50 T + 6123 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 44 T + 968 T^{2} - 7392 T^{3} - 18614593 T^{4} - 7392 p^{2} T^{5} + 968 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} + 82824 T^{3} - 38135518 T^{4} + 82824 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 36 T + 648 T^{2} + 51156 T^{3} - 41524018 T^{4} + 51156 p^{2} T^{5} + 648 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 32 T + 9282 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 140 T + 9800 T^{2} + 850080 T^{3} + 73070879 T^{4} + 850080 p^{2} T^{5} + 9800 p^{4} T^{6} + 140 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 84 T + 3528 T^{2} + 703164 T^{3} + 139944782 T^{4} + 703164 p^{2} T^{5} + 3528 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 128 T + 8192 T^{2} + 1245312 T^{3} + 189204482 T^{4} + 1245312 p^{2} T^{5} + 8192 p^{4} T^{6} + 128 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.55325993342280309463765997020, −7.27973989044219346570050027279, −6.93719757312355874655954538877, −6.84538033597799480299200491229, −6.71785943264832494704312472788, −6.15420136432246860445897393974, −6.00888480590822875194440996298, −5.99819511227004417705770916524, −5.42182577592978879724785581316, −5.15654216030803911754547595577, −5.15617924015385204166939833293, −4.35165409371747650314195927606, −4.35123709983302700715876781933, −4.31962325895068371636294695699, −3.99569383394204466482692748115, −3.48255051982399812458445132650, −3.23684897789566186612857225936, −3.04888388201779970463215319987, −2.44534624813439080510806869814, −1.89103383664320379668632683851, −1.74665506500339281933435954715, −1.38878928322988115976235505236, −1.01352165681998230817527628814, −1.00495312077349509364399048871, −0.72814288258402843999809142284,
0.72814288258402843999809142284, 1.00495312077349509364399048871, 1.01352165681998230817527628814, 1.38878928322988115976235505236, 1.74665506500339281933435954715, 1.89103383664320379668632683851, 2.44534624813439080510806869814, 3.04888388201779970463215319987, 3.23684897789566186612857225936, 3.48255051982399812458445132650, 3.99569383394204466482692748115, 4.31962325895068371636294695699, 4.35123709983302700715876781933, 4.35165409371747650314195927606, 5.15617924015385204166939833293, 5.15654216030803911754547595577, 5.42182577592978879724785581316, 5.99819511227004417705770916524, 6.00888480590822875194440996298, 6.15420136432246860445897393974, 6.71785943264832494704312472788, 6.84538033597799480299200491229, 6.93719757312355874655954538877, 7.27973989044219346570050027279, 7.55325993342280309463765997020
