| L(s) = 1 | − 4·2-s + 6·3-s + 8·4-s − 2·5-s − 24·6-s + 8·7-s − 8·8-s + 21·9-s + 8·10-s − 4·11-s + 48·12-s − 32·14-s − 12·15-s − 4·16-s + 12·17-s − 84·18-s + 12·19-s − 16·20-s + 48·21-s + 16·22-s − 48·24-s + 5·25-s + 54·27-s + 64·28-s − 18·29-s + 48·30-s + 6·31-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3.46·3-s + 4·4-s − 0.894·5-s − 9.79·6-s + 3.02·7-s − 2.82·8-s + 7·9-s + 2.52·10-s − 1.20·11-s + 13.8·12-s − 8.55·14-s − 3.09·15-s − 16-s + 2.91·17-s − 19.7·18-s + 2.75·19-s − 3.57·20-s + 10.4·21-s + 3.41·22-s − 9.79·24-s + 25-s + 10.3·27-s + 12.0·28-s − 3.34·29-s + 8.76·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.768398507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.768398507\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 12 T^{3} + 599 T^{4} + 12 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 9 T^{2} + 120 T^{3} - 244 T^{4} + 120 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 18 T + 189 T^{2} + 1458 T^{3} + 8852 T^{4} + 1458 p T^{5} + 189 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} + 108 T^{3} - 141 T^{4} + 108 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 372 T^{3} + 1886 T^{4} + 372 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 133 T^{2} + 1020 T^{3} + 7512 T^{4} + 1020 p T^{5} + 133 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 6 T + 90 T^{2} + 624 T^{3} + 4895 T^{4} + 624 p T^{5} + 90 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 390 T^{3} + 1256 T^{4} + 390 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 2914 T^{4} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 18 T + 204 T^{2} - 1728 T^{3} + 12107 T^{4} - 1728 p T^{5} + 204 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 45 T^{2} + 738 T^{3} - 4576 T^{4} + 738 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1380 T^{3} + 9506 T^{4} + 1380 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18 T + 284 T^{2} - 3168 T^{3} + 33267 T^{4} - 3168 p T^{5} + 284 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 28 T + 365 T^{2} - 2980 T^{3} + 23356 T^{4} - 2980 p T^{5} + 365 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 139 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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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
−8.413736471502459291830082599851, −8.002436547017668884173336226007, −7.973233732471883398177020575854, −7.73251914148829699345693159581, −7.72301093173146191366578498356, −7.33540747503466395117742305568, −7.23550534001813438367942115322, −7.21644743711787199819319418150, −6.93932457750440931013950468415, −6.04452704003901409032919803402, −5.74910235235652587430443592300, −5.12578769731310175283882111040, −4.99143709089572192984765956535, −4.87294444910155861493253679899, −4.77358111035545628538403890067, −3.94133861164625572821403634187, −3.80330049668994363816354288423, −3.25813430030778326209470169939, −3.16245773785974763026513044294, −3.10132213518760162573479705627, −2.20072315159733612027983961372, −1.92776377028690140961719067085, −1.65907284822874193602668733296, −1.45941311144212231409615495502, −1.01682826259216223595647889604,
1.01682826259216223595647889604, 1.45941311144212231409615495502, 1.65907284822874193602668733296, 1.92776377028690140961719067085, 2.20072315159733612027983961372, 3.10132213518760162573479705627, 3.16245773785974763026513044294, 3.25813430030778326209470169939, 3.80330049668994363816354288423, 3.94133861164625572821403634187, 4.77358111035545628538403890067, 4.87294444910155861493253679899, 4.99143709089572192984765956535, 5.12578769731310175283882111040, 5.74910235235652587430443592300, 6.04452704003901409032919803402, 6.93932457750440931013950468415, 7.21644743711787199819319418150, 7.23550534001813438367942115322, 7.33540747503466395117742305568, 7.72301093173146191366578498356, 7.73251914148829699345693159581, 7.973233732471883398177020575854, 8.002436547017668884173336226007, 8.413736471502459291830082599851
