| L(s) = 1 | + 8·5-s − 8·7-s + 12·13-s + 4·17-s − 8·23-s + 38·25-s − 64·35-s + 4·37-s + 16·41-s + 24·43-s + 32·49-s + 12·53-s + 24·59-s + 96·65-s + 8·67-s + 4·73-s + 8·79-s − 81-s + 32·83-s + 32·85-s − 96·91-s + 20·97-s − 8·101-s − 16·103-s − 8·107-s − 12·113-s − 64·115-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 3.02·7-s + 3.32·13-s + 0.970·17-s − 1.66·23-s + 38/5·25-s − 10.8·35-s + 0.657·37-s + 2.49·41-s + 3.65·43-s + 32/7·49-s + 1.64·53-s + 3.12·59-s + 11.9·65-s + 0.977·67-s + 0.468·73-s + 0.900·79-s − 1/9·81-s + 3.51·83-s + 3.47·85-s − 10.0·91-s + 2.03·97-s − 0.796·101-s − 1.57·103-s − 0.773·107-s − 1.12·113-s − 5.96·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.60616450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.60616450\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 324 T^{3} + 1262 T^{4} - 324 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 60 T^{3} + 446 T^{4} - 60 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 386 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1094 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 244 T^{3} - 2162 T^{4} + 244 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2376 T^{3} + 16466 T^{4} - 2376 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 6046 T^{4} - 660 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 472 T^{3} + 6898 T^{4} - 472 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4006 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 142 T^{2} + p^{2} T^{4} ) \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 6240 T^{3} + 63506 T^{4} - 6240 p T^{5} + 512 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 19334 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1660 T^{3} + 13582 T^{4} - 1660 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.37324679981544697867096924300, −6.34707358569238709955674619125, −6.13782428342623711810453523268, −6.07729195737725360518240194283, −5.83647083271117622003369209966, −5.44942128813725968819610734156, −5.42437255914835200099839650891, −5.35614084928178554928463384249, −5.29844189217233516570266329389, −4.36202610284005125097471416710, −4.25215672779272073111138638255, −4.13337577898750993258042309792, −3.96338357931677975704060342641, −3.49009516529616672467208103833, −3.39743269108983027259656642156, −3.21016238693102485309374361947, −2.95635346546357962155082297915, −2.41857985808949959039766521344, −2.32531473344134049623209543101, −2.19438948224497527073737064795, −2.13404369325961730368542105147, −1.37658194746318917254880932753, −0.963976741028657649575239368549, −0.834403957783126961050547028966, −0.830082995303393776179049750998,
0.830082995303393776179049750998, 0.834403957783126961050547028966, 0.963976741028657649575239368549, 1.37658194746318917254880932753, 2.13404369325961730368542105147, 2.19438948224497527073737064795, 2.32531473344134049623209543101, 2.41857985808949959039766521344, 2.95635346546357962155082297915, 3.21016238693102485309374361947, 3.39743269108983027259656642156, 3.49009516529616672467208103833, 3.96338357931677975704060342641, 4.13337577898750993258042309792, 4.25215672779272073111138638255, 4.36202610284005125097471416710, 5.29844189217233516570266329389, 5.35614084928178554928463384249, 5.42437255914835200099839650891, 5.44942128813725968819610734156, 5.83647083271117622003369209966, 6.07729195737725360518240194283, 6.13782428342623711810453523268, 6.34707358569238709955674619125, 6.37324679981544697867096924300
