| L(s) = 1 | + 4·2-s + 3·4-s + 18·5-s − 28·7-s + 72·10-s − 44·11-s − 134·13-s − 112·14-s − 25·16-s + 74·17-s − 164·19-s + 54·20-s − 176·22-s − 194·23-s − 69·25-s − 536·26-s − 84·28-s + 108·29-s − 412·31-s − 244·32-s + 296·34-s − 504·35-s + 286·37-s − 656·38-s + 18·41-s − 496·43-s − 132·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/8·4-s + 1.60·5-s − 1.51·7-s + 2.27·10-s − 1.20·11-s − 2.85·13-s − 2.13·14-s − 0.390·16-s + 1.05·17-s − 1.98·19-s + 0.603·20-s − 1.70·22-s − 1.75·23-s − 0.551·25-s − 4.04·26-s − 0.566·28-s + 0.691·29-s − 2.38·31-s − 1.34·32-s + 1.49·34-s − 2.43·35-s + 1.27·37-s − 2.80·38-s + 0.0685·41-s − 1.75·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - p^{2} T + 13 T^{2} - 5 p^{3} T^{3} + 73 p T^{4} - 5 p^{6} T^{5} + 13 p^{6} T^{6} - p^{11} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 18 T + 393 T^{2} - 758 p T^{3} + 56148 T^{4} - 758 p^{4} T^{5} + 393 p^{6} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 134 T + 10520 T^{2} + 559034 T^{3} + 26667070 T^{4} + 559034 p^{3} T^{5} + 10520 p^{6} T^{6} + 134 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 74 T + 17468 T^{2} - 831750 T^{3} + 118657126 T^{4} - 831750 p^{3} T^{5} + 17468 p^{6} T^{6} - 74 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 164 T + 11968 T^{2} + 138820 T^{3} - 15467058 T^{4} + 138820 p^{3} T^{5} + 11968 p^{6} T^{6} + 164 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 194 T + 46053 T^{2} + 5374478 T^{3} + 784861428 T^{4} + 5374478 p^{3} T^{5} + 46053 p^{6} T^{6} + 194 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 108 T + 42724 T^{2} + 1514908 T^{3} + 528463590 T^{4} + 1514908 p^{3} T^{5} + 42724 p^{6} T^{6} - 108 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 412 T + 51613 T^{2} - 11046366 T^{3} - 3907770320 T^{4} - 11046366 p^{3} T^{5} + 51613 p^{6} T^{6} + 412 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 286 T + 135513 T^{2} - 31935490 T^{3} + 9691467156 T^{4} - 31935490 p^{3} T^{5} + 135513 p^{6} T^{6} - 286 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 18 T + 114524 T^{2} + 18203666 T^{3} + 5875037446 T^{4} + 18203666 p^{3} T^{5} + 114524 p^{6} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 496 T + 308268 T^{2} + 103155312 T^{3} + 35378135478 T^{4} + 103155312 p^{3} T^{5} + 308268 p^{6} T^{6} + 496 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 62 T + 345832 T^{2} + 21522086 T^{3} + 50715680878 T^{4} + 21522086 p^{3} T^{5} + 345832 p^{6} T^{6} + 62 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 828 T + 705892 T^{2} - 360136820 T^{3} + 165458948486 T^{4} - 360136820 p^{3} T^{5} + 705892 p^{6} T^{6} - 828 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1224 T + 1275413 T^{2} - 804281630 T^{3} + 441199752912 T^{4} - 804281630 p^{3} T^{5} + 1275413 p^{6} T^{6} - 1224 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 350 T + 430588 T^{2} + 138974618 T^{3} + 88699741846 T^{4} + 138974618 p^{3} T^{5} + 430588 p^{6} T^{6} + 350 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 1498 T + 1714917 T^{2} + 1237137334 T^{3} + 793168291468 T^{4} + 1237137334 p^{3} T^{5} + 1714917 p^{6} T^{6} + 1498 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 2326 T + 3445789 T^{2} + 3267789594 T^{3} + 2320231641060 T^{4} + 3267789594 p^{3} T^{5} + 3445789 p^{6} T^{6} + 2326 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 1630 T + 2286292 T^{2} + 1913055042 T^{3} + 1441611662422 T^{4} + 1913055042 p^{3} T^{5} + 2286292 p^{6} T^{6} + 1630 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 1020 T + 1989936 T^{2} + 1295128300 T^{3} + 1427479601566 T^{4} + 1295128300 p^{3} T^{5} + 1989936 p^{6} T^{6} + 1020 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1920 T + 2662892 T^{2} - 2752584832 T^{3} + 2433754978902 T^{4} - 2752584832 p^{3} T^{5} + 2662892 p^{6} T^{6} - 1920 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 1550 T + 1862281 T^{2} + 874387910 T^{3} + 705085008356 T^{4} + 874387910 p^{3} T^{5} + 1862281 p^{6} T^{6} + 1550 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2202 T + 5018625 T^{2} + 6116849198 T^{3} + 7416588769820 T^{4} + 6116849198 p^{3} T^{5} + 5018625 p^{6} T^{6} + 2202 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−7.43908560853462589685585361744, −7.42069251850984945151730663946, −7.02836232436908219879105323332, −6.88384612999257518466981169394, −6.48999357559035646455439210864, −6.25607465360071010486670127776, −5.97908099531001712952082097493, −5.83209613863528777573237939348, −5.72566311713299537455326859646, −5.37709473805517550820976300062, −5.27849320457487709593338216840, −4.96037070827037694424298004050, −4.77636070803062740776977182768, −4.26722120098286793240652316473, −4.17270610490476072193330805663, −4.01275152710589195220119942013, −3.89631367850701075533333192742, −3.20097051193696083071281728439, −2.89011072785254005961638972133, −2.87829508660891421682969294003, −2.27790683984819339740675165412, −2.23792866836422627527643145512, −2.16301159074055280804543063375, −1.48526244251488619030461131691, −1.34468543103861334978274148012, 0, 0, 0, 0,
1.34468543103861334978274148012, 1.48526244251488619030461131691, 2.16301159074055280804543063375, 2.23792866836422627527643145512, 2.27790683984819339740675165412, 2.87829508660891421682969294003, 2.89011072785254005961638972133, 3.20097051193696083071281728439, 3.89631367850701075533333192742, 4.01275152710589195220119942013, 4.17270610490476072193330805663, 4.26722120098286793240652316473, 4.77636070803062740776977182768, 4.96037070827037694424298004050, 5.27849320457487709593338216840, 5.37709473805517550820976300062, 5.72566311713299537455326859646, 5.83209613863528777573237939348, 5.97908099531001712952082097493, 6.25607465360071010486670127776, 6.48999357559035646455439210864, 6.88384612999257518466981169394, 7.02836232436908219879105323332, 7.42069251850984945151730663946, 7.43908560853462589685585361744
