| L(s) = 1 | + 4·5-s + 24·7-s + 56·11-s − 24·13-s + 12·17-s + 16·23-s + 16·25-s − 44·31-s + 96·35-s − 68·37-s + 16·41-s + 28·43-s − 16·47-s + 288·49-s − 12·53-s + 224·55-s + 84·61-s − 96·65-s − 116·67-s + 200·71-s + 108·73-s + 1.34e3·77-s + 200·83-s + 48·85-s − 576·91-s + 176·97-s + 512·101-s + ⋯ |
| L(s) = 1 | + 4/5·5-s + 24/7·7-s + 5.09·11-s − 1.84·13-s + 0.705·17-s + 0.695·23-s + 0.639·25-s − 1.41·31-s + 2.74·35-s − 1.83·37-s + 0.390·41-s + 0.651·43-s − 0.340·47-s + 5.87·49-s − 0.226·53-s + 4.07·55-s + 1.37·61-s − 1.47·65-s − 1.73·67-s + 2.81·71-s + 1.47·73-s + 17.4·77-s + 2.40·83-s + 0.564·85-s − 6.32·91-s + 1.81·97-s + 5.06·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(28.74163779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(28.74163779\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2616 T^{3} + 20162 T^{4} - 2616 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 28 T + 432 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 5496 T^{3} + 101282 T^{4} + 5496 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 2784 T^{3} + 104399 T^{4} - 2784 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1094 T^{2} + 533715 T^{4} - 1094 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} + 1824 T^{3} - 387457 T^{4} + 1824 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1198 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 22 T + 2037 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 68 T + 2312 T^{2} + 14892 T^{3} - 1226578 T^{4} + 14892 p^{2} T^{5} + 2312 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 3228 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 28 T + 392 T^{2} - 49140 T^{3} + 6151214 T^{4} - 49140 p^{2} T^{5} + 392 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} - 19632 T^{3} - 8795038 T^{4} - 19632 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 3648 T^{3} - 6090193 T^{4} + 3648 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 13100 T^{2} + 66967878 T^{4} - 13100 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 42 T + 7787 T^{2} - 42 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 116 T + 6728 T^{2} + 547404 T^{3} + 44485022 T^{4} + 547404 p^{2} T^{5} + 6728 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 100 T + 10848 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 108 T + 5832 T^{2} + 77004 T^{3} - 35489026 T^{4} + 77004 p^{2} T^{5} + 5832 p^{4} T^{6} - 108 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 10406 T^{2} + 103410771 T^{4} - 10406 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 200 T + 20000 T^{2} - 937200 T^{3} + 39063983 T^{4} - 937200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16280 T^{2} + 147092178 T^{4} - 16280 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 176 T + 15488 T^{2} - 1653168 T^{3} + 176456642 T^{4} - 1653168 p^{2} T^{5} + 15488 p^{4} T^{6} - 176 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.03449837911374670681535847361, −6.57444383157452450868137171353, −6.31767810490728641407609281072, −6.25243670075037919863612581640, −6.19951147313447166012627680578, −5.56793553470093798074138998178, −5.48454923601609309752591875941, −5.11825524684371407345714480940, −5.10845600949535281512911385775, −4.76916773136825669937836139866, −4.51044775221910934917804844774, −4.43823612008109642071527813229, −4.27823543184819154452720457008, −3.62275066228114792071075578288, −3.61574793741171836473117042606, −3.45652098297814568116854404882, −3.23763225973823049349649545146, −2.40577501730962454400282491725, −2.01131057254491557395166226021, −1.91537688109506284348224403864, −1.90236359876800609402468413775, −1.64355569206993600129596248068, −0.940625622067311762235521572923, −0.905289588989207485113149220126, −0.812485657898713177650571247512,
0.812485657898713177650571247512, 0.905289588989207485113149220126, 0.940625622067311762235521572923, 1.64355569206993600129596248068, 1.90236359876800609402468413775, 1.91537688109506284348224403864, 2.01131057254491557395166226021, 2.40577501730962454400282491725, 3.23763225973823049349649545146, 3.45652098297814568116854404882, 3.61574793741171836473117042606, 3.62275066228114792071075578288, 4.27823543184819154452720457008, 4.43823612008109642071527813229, 4.51044775221910934917804844774, 4.76916773136825669937836139866, 5.10845600949535281512911385775, 5.11825524684371407345714480940, 5.48454923601609309752591875941, 5.56793553470093798074138998178, 6.19951147313447166012627680578, 6.25243670075037919863612581640, 6.31767810490728641407609281072, 6.57444383157452450868137171353, 7.03449837911374670681535847361
