| L(s) = 1 | − 2·3-s − 10·5-s − 22·7-s + 37·9-s + 28·11-s − 72·13-s + 20·15-s + 76·17-s + 160·19-s + 44·21-s − 22·23-s + 25·25-s − 106·27-s − 500·29-s − 132·31-s − 56·33-s + 220·35-s + 416·37-s + 144·39-s − 212·41-s + 1.33e3·43-s − 370·45-s − 196·47-s + 127·49-s − 152·51-s + 952·53-s − 280·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.894·5-s − 1.18·7-s + 1.37·9-s + 0.767·11-s − 1.53·13-s + 0.344·15-s + 1.08·17-s + 1.93·19-s + 0.457·21-s − 0.199·23-s + 1/5·25-s − 0.755·27-s − 3.20·29-s − 0.764·31-s − 0.295·33-s + 1.06·35-s + 1.84·37-s + 0.591·39-s − 0.807·41-s + 4.72·43-s − 1.22·45-s − 0.608·47-s + 0.370·49-s − 0.417·51-s + 2.46·53-s − 0.686·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3218805237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3218805237\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 22 T + 51 p T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - 11 p T^{2} - 34 T^{3} + 532 T^{4} - 34 p^{3} T^{5} - 11 p^{7} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 28 T - 1874 T^{2} + 112 T^{3} + 4249899 T^{4} + 112 p^{3} T^{5} - 1874 p^{6} T^{6} - 28 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 36 T + 4518 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 76 T + 338 T^{2} + 333488 T^{3} - 22943213 T^{4} + 333488 p^{3} T^{5} + 338 p^{6} T^{6} - 76 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 160 T + 6834 T^{2} - 807680 T^{3} + 129526475 T^{4} - 807680 p^{3} T^{5} + 6834 p^{6} T^{6} - 160 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 22 T + 3407 T^{2} - 599654 T^{3} - 145380788 T^{4} - 599654 p^{3} T^{5} + 3407 p^{6} T^{6} + 22 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 250 T + 57203 T^{2} + 250 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 132 T - 26514 T^{2} - 2065008 T^{3} + 523965779 T^{4} - 2065008 p^{3} T^{5} - 26514 p^{6} T^{6} + 132 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 416 T + 65478 T^{2} - 2609152 T^{3} + 241494107 T^{4} - 2609152 p^{3} T^{5} + 65478 p^{6} T^{6} - 416 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 106 T + 138851 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 666 T + 269853 T^{2} - 666 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 196 T - 170642 T^{2} + 276752 T^{3} + 28937567667 T^{4} + 276752 p^{3} T^{5} - 170642 p^{6} T^{6} + 196 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 952 T + 422302 T^{2} - 177308096 T^{3} + 77165754267 T^{4} - 177308096 p^{3} T^{5} + 422302 p^{6} T^{6} - 952 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 840 T + 4406 p T^{2} - 29305920 T^{3} + 11504401275 T^{4} - 29305920 p^{3} T^{5} + 4406 p^{7} T^{6} - 840 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 98 T + 203041 T^{2} - 63445102 T^{3} - 16282426916 T^{4} - 63445102 p^{3} T^{5} + 203041 p^{6} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 1286 T + 648343 T^{2} + 519450122 T^{3} + 423076706692 T^{4} + 519450122 p^{3} T^{5} + 648343 p^{6} T^{6} + 1286 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1064 T + 753846 T^{2} + 1064 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 172 T - 727998 T^{2} + 3517744 T^{3} + 411087581507 T^{4} + 3517744 p^{3} T^{5} - 727998 p^{6} T^{6} - 172 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1240 T + 258714 T^{2} + 363081920 T^{3} + 634365179075 T^{4} + 363081920 p^{3} T^{5} + 258714 p^{6} T^{6} + 1240 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1906 T + 2051733 T^{2} - 1906 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 650 T - 883111 T^{2} - 67812550 T^{3} + 909789389860 T^{4} - 67812550 p^{3} T^{5} - 883111 p^{6} T^{6} + 650 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 628 T + 1423942 T^{2} + 628 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.42629454442602882876776506856, −7.28234222843529939187370168810, −7.00644164460057829223379790461, −6.81198922741125223281739790459, −6.16742403933160630788682080132, −6.13878450838183838088913039831, −5.87215423299870552596009953274, −5.66870022061101161209630455894, −5.40277522088290810413145152388, −5.22594755363936909247889558124, −4.84114406156938247556999740442, −4.41034490168100057354623885745, −4.23136207829673363333711323833, −4.06803820263160863717112323367, −3.65394260036677670866068763202, −3.62142364646358839479460412987, −3.36129634130138554717922690863, −2.67506548800961787280250516411, −2.59909799691700195393692563077, −2.38196961943913861446264964830, −1.58757368934471895175114924037, −1.49428960467761865491889207984, −0.909220349950801819471212852298, −0.73812893630720000599622993380, −0.098376892327807487525530076972,
0.098376892327807487525530076972, 0.73812893630720000599622993380, 0.909220349950801819471212852298, 1.49428960467761865491889207984, 1.58757368934471895175114924037, 2.38196961943913861446264964830, 2.59909799691700195393692563077, 2.67506548800961787280250516411, 3.36129634130138554717922690863, 3.62142364646358839479460412987, 3.65394260036677670866068763202, 4.06803820263160863717112323367, 4.23136207829673363333711323833, 4.41034490168100057354623885745, 4.84114406156938247556999740442, 5.22594755363936909247889558124, 5.40277522088290810413145152388, 5.66870022061101161209630455894, 5.87215423299870552596009953274, 6.13878450838183838088913039831, 6.16742403933160630788682080132, 6.81198922741125223281739790459, 7.00644164460057829223379790461, 7.28234222843529939187370168810, 7.42629454442602882876776506856
