| L(s) = 1 | − 10·2-s + 50·4-s − 550·7-s + 240·8-s + 1.05e3·11-s − 1.96e3·13-s + 5.50e3·14-s − 708·16-s − 3.28e3·17-s − 1.05e4·22-s − 3.90e4·23-s + 1.96e4·26-s − 2.75e4·28-s − 3.31e4·31-s − 4.34e4·32-s + 3.28e4·34-s − 1.46e5·37-s + 2.13e5·41-s + 7.20e4·43-s + 5.26e4·44-s + 3.90e5·46-s + 830·47-s + 1.51e5·49-s − 9.80e4·52-s − 2.96e4·53-s − 1.32e5·56-s − 1.11e5·61-s + ⋯ |
| L(s) = 1 | − 5/4·2-s + 0.781·4-s − 1.60·7-s + 0.468·8-s + 0.790·11-s − 0.892·13-s + 2.00·14-s − 0.172·16-s − 0.667·17-s − 0.987·22-s − 3.20·23-s + 1.11·26-s − 1.25·28-s − 1.11·31-s − 1.32·32-s + 0.834·34-s − 2.89·37-s + 3.10·41-s + 0.906·43-s + 0.617·44-s + 4.00·46-s + 0.00799·47-s + 1.28·49-s − 0.696·52-s − 0.198·53-s − 0.751·56-s − 0.489·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8411266174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8411266174\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 5 p T + 25 p T^{2} - 15 p^{4} T^{3} - 103 p^{6} T^{4} - 15 p^{10} T^{5} + 25 p^{13} T^{6} + 5 p^{19} T^{7} + p^{24} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 550 T + 151250 T^{2} + 78815550 T^{3} + 40412328098 T^{4} + 78815550 p^{6} T^{5} + 151250 p^{12} T^{6} + 550 p^{18} T^{7} + p^{24} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 526 T + 225606 p T^{2} - 526 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 1960 T + 1920800 T^{2} + 6864764760 T^{3} + 22780068732638 T^{4} + 6864764760 p^{6} T^{5} + 1920800 p^{12} T^{6} + 1960 p^{18} T^{7} + p^{24} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3280 T + 5379200 T^{2} + 52715999280 T^{3} + 451561024170878 T^{4} + 52715999280 p^{6} T^{5} + 5379200 p^{12} T^{6} + 3280 p^{18} T^{7} + p^{24} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 55109524 T^{2} + 4864046077848966 T^{4} - 55109524 p^{12} T^{6} + p^{24} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 39010 T + 760890050 T^{2} + 12796505733210 T^{3} + 182810834786595458 T^{4} + 12796505733210 p^{6} T^{5} + 760890050 p^{12} T^{6} + 39010 p^{18} T^{7} + p^{24} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1657037284 T^{2} + 1284148562793102246 T^{4} - 1657037284 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 16586 T + 1245680586 T^{2} + 16586 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 146860 T + 10783929800 T^{2} + 657351006913860 T^{3} + 36420548331850749038 T^{4} + 657351006913860 p^{6} T^{5} + 10783929800 p^{12} T^{6} + 146860 p^{18} T^{7} + p^{24} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 106966 T + 7285264146 T^{2} - 106966 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 72050 T + 2595601250 T^{2} - 482755757677050 T^{3} + 89644137077771169698 T^{4} - 482755757677050 p^{6} T^{5} + 2595601250 p^{12} T^{6} - 72050 p^{18} T^{7} + p^{24} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 830 T + 344450 T^{2} - 6853931089830 T^{3} + \)\(13\!\cdots\!18\)\( T^{4} - 6853931089830 p^{6} T^{5} + 344450 p^{12} T^{6} - 830 p^{18} T^{7} + p^{24} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 29620 T + 438672200 T^{2} + 493381118739420 T^{3} + \)\(52\!\cdots\!18\)\( T^{4} + 493381118739420 p^{6} T^{5} + 438672200 p^{12} T^{6} + 29620 p^{18} T^{7} + p^{24} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 166570372564 T^{2} + \)\(10\!\cdots\!86\)\( T^{4} - 166570372564 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 55526 T + 101522017266 T^{2} + 55526 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 146930 T + 10794212450 T^{2} - 2898062262927930 T^{3} - \)\(42\!\cdots\!22\)\( T^{4} - 2898062262927930 p^{6} T^{5} + 10794212450 p^{12} T^{6} - 146930 p^{18} T^{7} + p^{24} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 655094 T + 342881964426 T^{2} + 655094 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 553540 T + 153203265800 T^{2} + 75685034633171340 T^{3} + \)\(37\!\cdots\!58\)\( T^{4} + 75685034633171340 p^{6} T^{5} + 153203265800 p^{12} T^{6} + 553540 p^{18} T^{7} + p^{24} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 713041150084 T^{2} + \)\(23\!\cdots\!46\)\( T^{4} - 713041150084 p^{12} T^{6} + p^{24} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 536870 T + 144114698450 T^{2} + 4448869644102930 T^{3} - \)\(11\!\cdots\!22\)\( T^{4} + 4448869644102930 p^{6} T^{5} + 144114698450 p^{12} T^{6} - 536870 p^{18} T^{7} + p^{24} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1229834859844 T^{2} + \)\(77\!\cdots\!26\)\( T^{4} - 1229834859844 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 59420 T + 1765368200 T^{2} - 49275595300926420 T^{3} + \)\(13\!\cdots\!18\)\( T^{4} - 49275595300926420 p^{6} T^{5} + 1765368200 p^{12} T^{6} - 59420 p^{18} T^{7} + p^{24} T^{8} \) |
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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
−7.65872376348763521438468307292, −7.55763428593886972294070703424, −7.31591066909716110491955492932, −7.28371763756346260519861920798, −6.80152625897442549464482289595, −6.55932115407714649215645439285, −6.26971658841026198970377582427, −5.84370734176336868912970685488, −5.73886154395665208290482685062, −5.66684623409682604010779428373, −5.11733752694507442423048789125, −4.53776550631927615079182563176, −4.42167101056894620546951355515, −3.96267564426709079191321552730, −3.82615742582303713254523131735, −3.61156495581352128298256074014, −2.96010640458513502852509501272, −2.82444682452276424122949174019, −2.39511826875919099827228547707, −1.71572227568904441502434324162, −1.68736290166468712641034278233, −1.67386011247724231860254456281, −0.71675983729113858320466228679, −0.39705472159874535373290725341, −0.28693515262326200618502082219,
0.28693515262326200618502082219, 0.39705472159874535373290725341, 0.71675983729113858320466228679, 1.67386011247724231860254456281, 1.68736290166468712641034278233, 1.71572227568904441502434324162, 2.39511826875919099827228547707, 2.82444682452276424122949174019, 2.96010640458513502852509501272, 3.61156495581352128298256074014, 3.82615742582303713254523131735, 3.96267564426709079191321552730, 4.42167101056894620546951355515, 4.53776550631927615079182563176, 5.11733752694507442423048789125, 5.66684623409682604010779428373, 5.73886154395665208290482685062, 5.84370734176336868912970685488, 6.26971658841026198970377582427, 6.55932115407714649215645439285, 6.80152625897442549464482289595, 7.28371763756346260519861920798, 7.31591066909716110491955492932, 7.55763428593886972294070703424, 7.65872376348763521438468307292
