| L(s) = 1 | − 4·2-s + 8·4-s + 12·7-s − 4·8-s + 14·9-s − 4·11-s + 40·13-s − 48·14-s − 25·16-s + 12·17-s − 56·18-s + 16·22-s − 72·23-s − 160·26-s + 96·28-s − 40·29-s + 40·31-s + 56·32-s − 48·34-s + 112·36-s − 40·37-s + 32·41-s + 84·43-s − 32·44-s + 288·46-s + 4·47-s + 72·49-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·4-s + 12/7·7-s − 1/2·8-s + 14/9·9-s − 0.363·11-s + 3.07·13-s − 3.42·14-s − 1.56·16-s + 0.705·17-s − 3.11·18-s + 8/11·22-s − 3.13·23-s − 6.15·26-s + 24/7·28-s − 1.37·29-s + 1.29·31-s + 7/4·32-s − 1.41·34-s + 28/9·36-s − 1.08·37-s + 0.780·41-s + 1.95·43-s − 0.727·44-s + 6.26·46-s + 4/47·47-s + 1.46·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.314757341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.314757341\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 40 T + 56 p T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + p^{3} T^{2} + p^{2} T^{3} - 7 T^{4} + p^{4} T^{5} + p^{7} T^{6} + p^{8} T^{7} + p^{8} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 14 T^{2} + 19 p^{2} T^{4} - 14 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 744 T^{3} + 7519 T^{4} - 744 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 172 T^{3} - 2386 T^{4} + 172 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 227 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 232162 T^{4} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 36 T + 1292 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 20 T + 1532 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 40 T + 800 T^{2} - 39240 T^{3} + 1924322 T^{4} - 39240 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 46560 T^{3} + 2667767 T^{4} + 46560 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 57248 T^{3} + 6389378 T^{4} - 57248 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 42 T + 4049 T^{2} - 42 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 1736 T^{3} - 6608737 T^{4} + 1736 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 9936 T^{2} + 40061986 T^{4} - 9936 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 56 T + 1568 T^{2} - 2632 T^{3} - 12442366 T^{4} - 2632 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 148 T + 12878 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 84 T + 3528 T^{2} - 155316 T^{3} - 33332642 T^{4} - 155316 p^{2} T^{5} + 3528 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 4 p T + 8 p^{2} T^{2} - 57112 p T^{3} + 322400399 T^{4} - 57112 p^{3} T^{5} + 8 p^{6} T^{6} - 4 p^{7} T^{7} + p^{8} T^{8} \) |
| 73 | $C_2^3$ | \( 1 + 18164482 T^{4} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 32 T + 11528 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 52 T + 1352 T^{2} - 271804 T^{3} + 51880814 T^{4} - 271804 p^{2} T^{5} + 1352 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 200 T + 20000 T^{2} + 1908200 T^{3} + 179436962 T^{4} + 1908200 p^{2} T^{5} + 20000 p^{4} T^{6} + 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 68 T + 2312 T^{2} - 801924 T^{3} - 171375106 T^{4} - 801924 p^{2} T^{5} + 2312 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−8.144770335032713647913929852862, −8.016905406112331786512716643820, −7.68545322137077410458874602748, −7.63942423710149255377880345169, −7.55371564929208117540830208886, −7.30192319859980265561602506186, −6.43573605748402645022649447392, −6.43534237697754539157385127606, −6.43007937863455397305409905465, −6.11160778930969571347929896738, −5.41500135240155169501920733025, −5.34965082427199237398574372887, −5.28841559321870197173319778209, −4.46220396126474884131376699683, −4.24083917544227955738840978489, −4.16832074128564516659765546507, −3.79693619295793501480559776011, −3.72300671579341653209328351510, −2.83514370441348119857828970833, −2.55379930448024738087043661609, −1.76971777297774805106415997999, −1.64579660062231377149241233819, −1.39660477751098941016906898774, −1.28281630098897642222015603010, −0.37753564291355555324357618221,
0.37753564291355555324357618221, 1.28281630098897642222015603010, 1.39660477751098941016906898774, 1.64579660062231377149241233819, 1.76971777297774805106415997999, 2.55379930448024738087043661609, 2.83514370441348119857828970833, 3.72300671579341653209328351510, 3.79693619295793501480559776011, 4.16832074128564516659765546507, 4.24083917544227955738840978489, 4.46220396126474884131376699683, 5.28841559321870197173319778209, 5.34965082427199237398574372887, 5.41500135240155169501920733025, 6.11160778930969571347929896738, 6.43007937863455397305409905465, 6.43534237697754539157385127606, 6.43573605748402645022649447392, 7.30192319859980265561602506186, 7.55371564929208117540830208886, 7.63942423710149255377880345169, 7.68545322137077410458874602748, 8.016905406112331786512716643820, 8.144770335032713647913929852862
