| L(s) = 1 | + 16·2-s − 54·3-s + 64·4-s − 540·5-s − 864·6-s − 1.02e3·8-s + 729·9-s − 8.64e3·10-s − 1.19e3·11-s − 3.45e3·12-s − 2.46e4·13-s + 2.91e4·15-s − 1.63e4·16-s − 2.94e4·17-s + 1.16e4·18-s + 4.70e4·19-s − 3.45e4·20-s − 1.91e4·22-s + 9.40e4·23-s + 5.52e4·24-s + 2.26e5·25-s − 3.93e5·26-s + 3.93e4·27-s − 3.41e5·29-s + 4.66e5·30-s + 2.58e5·31-s − 6.55e4·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.93·5-s − 1.63·6-s − 0.707·8-s + 1/3·9-s − 2.73·10-s − 0.270·11-s − 0.577·12-s − 3.10·13-s + 2.23·15-s − 16-s − 1.45·17-s + 0.471·18-s + 1.57·19-s − 0.965·20-s − 0.383·22-s + 1.61·23-s + 0.816·24-s + 2.90·25-s − 4.39·26-s + 0.384·27-s − 2.59·29-s + 3.15·30-s + 1.56·31-s − 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.01721302053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01721302053\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p^{3} T + p^{6} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 108 p T + 2596 p^{2} T^{2} + 304344 p^{3} T^{3} + 36012999 p^{4} T^{4} + 304344 p^{10} T^{5} + 2596 p^{16} T^{6} + 108 p^{22} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 1196 T - 27613882 T^{2} - 11876332624 T^{3} + 435946651536427 T^{4} - 11876332624 p^{7} T^{5} - 27613882 p^{14} T^{6} + 1196 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 12312 T + 99570968 T^{2} + 12312 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 29484 T + 322588204 T^{2} - 8077415824296 T^{3} - 226187173694364513 T^{4} - 8077415824296 p^{7} T^{5} + 322588204 p^{14} T^{6} + 29484 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 47088 T + 902711442 T^{2} + 22280872687488 T^{3} - 1052880565778475733 T^{4} + 22280872687488 p^{7} T^{5} + 902711442 p^{14} T^{6} - 47088 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 94052 T + 1475279102 T^{2} - 52748942684816 T^{3} + 12613095628633527907 T^{4} - 52748942684816 p^{7} T^{5} + 1475279102 p^{14} T^{6} - 94052 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 170600 T + 41693255666 T^{2} + 170600 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 258984 T - 4480363142 T^{2} - 4280447973029184 T^{3} + \)\(25\!\cdots\!67\)\( T^{4} - 4280447973029184 p^{7} T^{5} - 4480363142 p^{14} T^{6} - 258984 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 648856 T + 135997011478 T^{2} - 61740817520417152 T^{3} + \)\(32\!\cdots\!87\)\( T^{4} - 61740817520417152 p^{7} T^{5} + 135997011478 p^{14} T^{6} - 648856 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 605556 T + 416076703796 T^{2} + 605556 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 698480 T + 535788255846 T^{2} + 698480 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 816912 T - 486667751318 T^{2} + 114994027781212032 T^{3} + \)\(73\!\cdots\!03\)\( T^{4} + 114994027781212032 p^{7} T^{5} - 486667751318 p^{14} T^{6} + 816912 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1923508 T + 481825150322 T^{2} - 1670827518121566544 T^{3} + \)\(46\!\cdots\!27\)\( T^{4} - 1670827518121566544 p^{7} T^{5} + 481825150322 p^{14} T^{6} - 1923508 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 81864 T - 2292506761454 T^{2} + 219239527631274432 T^{3} - \)\(92\!\cdots\!85\)\( T^{4} + 219239527631274432 p^{7} T^{5} - 2292506761454 p^{14} T^{6} - 81864 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1616544 T - 1741805663448 T^{2} - 3120682428761861952 T^{3} + \)\(87\!\cdots\!59\)\( T^{4} - 3120682428761861952 p^{7} T^{5} - 1741805663448 p^{14} T^{6} + 1616544 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 5424 T - 2686337401526 T^{2} + 51175745855801856 T^{3} - \)\(29\!\cdots\!73\)\( T^{4} + 51175745855801856 p^{7} T^{5} - 2686337401526 p^{14} T^{6} - 5424 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 2492964 T + 2402383804306 T^{2} + 2492964 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4710744 T - 4712548213920 T^{2} + 22653309377619710928 T^{3} + \)\(35\!\cdots\!35\)\( T^{4} + 22653309377619710928 p^{7} T^{5} - 4712548213920 p^{14} T^{6} + 4710744 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 3416656 T - 16415423029118 T^{2} - 35255983728480310784 T^{3} + \)\(19\!\cdots\!11\)\( T^{4} - 35255983728480310784 p^{7} T^{5} - 16415423029118 p^{14} T^{6} + 3416656 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 8921448 T + 74139859271942 T^{2} - 8921448 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12374532 T + 36847841483068 T^{2} + \)\(34\!\cdots\!36\)\( T^{3} + \)\(47\!\cdots\!99\)\( T^{4} + \)\(34\!\cdots\!36\)\( p^{7} T^{5} + 36847841483068 p^{14} T^{6} + 12374532 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 11296584 T + 159222144655440 T^{2} + 11296584 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−7.19372717166763524443282543548, −7.00354555392393071973063247600, −6.96071209148334155809625143481, −6.48274049469276865503516575472, −6.19387862859329543480327377574, −5.96555986252891626465538329424, −5.41341351146950740890345396633, −5.27800530453057311643096538219, −5.07105035655427737807322838605, −4.83396623303813100283414273479, −4.67450372833506227838485553143, −4.63088673542699286657070684565, −4.22361973486608987298016253571, −3.67785906461735388962655685822, −3.64922001297478728360651975081, −3.20604284257515557751206280502, −2.96683272339681349683729663882, −2.65826338405347393626656597972, −2.50631735738321991031330012456, −1.99456571732652508579446822834, −1.44784042333171460711734877705, −1.12791331367024790688645648651, −0.71555204598066805016290251408, −0.13933595057218425724855472398, −0.06201554794410873072722200407,
0.06201554794410873072722200407, 0.13933595057218425724855472398, 0.71555204598066805016290251408, 1.12791331367024790688645648651, 1.44784042333171460711734877705, 1.99456571732652508579446822834, 2.50631735738321991031330012456, 2.65826338405347393626656597972, 2.96683272339681349683729663882, 3.20604284257515557751206280502, 3.64922001297478728360651975081, 3.67785906461735388962655685822, 4.22361973486608987298016253571, 4.63088673542699286657070684565, 4.67450372833506227838485553143, 4.83396623303813100283414273479, 5.07105035655427737807322838605, 5.27800530453057311643096538219, 5.41341351146950740890345396633, 5.96555986252891626465538329424, 6.19387862859329543480327377574, 6.48274049469276865503516575472, 6.96071209148334155809625143481, 7.00354555392393071973063247600, 7.19372717166763524443282543548
