| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 4·5-s − 4·6-s + 6·7-s + 2·8-s + 4·9-s + 8·10-s − 6·11-s + 2·12-s − 12·14-s − 8·15-s − 4·16-s + 6·17-s − 8·18-s − 6·19-s − 4·20-s + 12·21-s + 12·22-s + 12·23-s + 4·24-s + 10·25-s + 4·27-s + 6·28-s + 12·29-s + 16·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 1.78·5-s − 1.63·6-s + 2.26·7-s + 0.707·8-s + 4/3·9-s + 2.52·10-s − 1.80·11-s + 0.577·12-s − 3.20·14-s − 2.06·15-s − 16-s + 1.45·17-s − 1.88·18-s − 1.37·19-s − 0.894·20-s + 2.61·21-s + 2.55·22-s + 2.50·23-s + 0.816·24-s + 2·25-s + 0.769·27-s + 1.13·28-s + 2.22·29-s + 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.937597738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.937597738\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T + 4 T^{3} - 5 T^{4} + 4 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 20 T^{2} + 108 T^{3} - 645 T^{4} + 108 p T^{5} + 20 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + T^{2} - 18 T^{3} + 180 T^{4} - 18 p T^{5} + p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 74 T^{2} - 288 T^{3} + 1059 T^{4} - 288 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 12 T + 62 T^{2} - 288 T^{3} + 1707 T^{4} - 288 p T^{5} + 62 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 61 T^{2} - 108 T^{3} + 192 T^{4} - 108 p T^{5} + 61 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 26 T^{2} - 1005 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 10 T^{2} - 3381 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T - 92 T^{2} - 52 T^{3} + 5251 T^{4} - 52 p T^{5} - 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 6 T + 80 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 132 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 24 T + 281 T^{2} + 2808 T^{3} + 27840 T^{4} + 2808 p T^{5} + 281 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 6 T - 20 T^{2} + 828 T^{3} - 9765 T^{4} + 828 p T^{5} - 20 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.93394612366929743507222958328, −6.51194614666180290725838577816, −6.31978851385086185502573350421, −6.29254301033284458058803726679, −5.62867957331346145115317194941, −5.57545999904518620816433855805, −5.19132636511584855336809918458, −4.88725846257365959316905017923, −4.84253105455964250925700708913, −4.76062182357313423418662933609, −4.35104266253413713642644959802, −4.30457352042420261228206353138, −4.25096091262461337598262828267, −3.64756303207757472384509125414, −3.53208252165982628395605997912, −3.01904704863826243820153448571, −2.90229547069996658656909013299, −2.70069514871009343499847631377, −2.53914795657387758308457569853, −2.21776201677307074824973015224, −1.51844330631269203433712910445, −1.33327784754200307017190985252, −1.32657033784862986921348477638, −0.61440805282310129172825457366, −0.58730806193021268247447899331,
0.58730806193021268247447899331, 0.61440805282310129172825457366, 1.32657033784862986921348477638, 1.33327784754200307017190985252, 1.51844330631269203433712910445, 2.21776201677307074824973015224, 2.53914795657387758308457569853, 2.70069514871009343499847631377, 2.90229547069996658656909013299, 3.01904704863826243820153448571, 3.53208252165982628395605997912, 3.64756303207757472384509125414, 4.25096091262461337598262828267, 4.30457352042420261228206353138, 4.35104266253413713642644959802, 4.76062182357313423418662933609, 4.84253105455964250925700708913, 4.88725846257365959316905017923, 5.19132636511584855336809918458, 5.57545999904518620816433855805, 5.62867957331346145115317194941, 6.29254301033284458058803726679, 6.31978851385086185502573350421, 6.51194614666180290725838577816, 6.93394612366929743507222958328
