| L(s) = 1 | − 2·2-s + 6·3-s + 4-s − 3·5-s − 12·6-s − 3·7-s + 2·8-s + 21·9-s + 6·10-s − 5·11-s + 6·12-s + 6·14-s − 18·15-s − 4·16-s − 17-s − 42·18-s − 15·19-s − 3·20-s − 18·21-s + 10·22-s − 3·23-s + 12·24-s + 54·27-s − 3·28-s − 4·29-s + 36·30-s + 13·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3.46·3-s + 1/2·4-s − 1.34·5-s − 4.89·6-s − 1.13·7-s + 0.707·8-s + 7·9-s + 1.89·10-s − 1.50·11-s + 1.73·12-s + 1.60·14-s − 4.64·15-s − 16-s − 0.242·17-s − 9.89·18-s − 3.44·19-s − 0.670·20-s − 3.92·21-s + 2.13·22-s − 0.625·23-s + 2.44·24-s + 10.3·27-s − 0.566·28-s − 0.742·29-s + 6.57·30-s + 2.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2023810770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2023810770\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + T - 19 T^{2} - 14 T^{3} + 94 T^{4} - 14 p T^{5} - 19 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 15 T + 127 T^{2} + 780 T^{3} + 3768 T^{4} + 780 p T^{5} + 127 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 3 T + 7 T^{2} + 12 T^{3} - 444 T^{4} + 12 p T^{5} + 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 13 T + 79 T^{2} - 364 T^{3} + 1900 T^{4} - 364 p T^{5} + 79 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 7 T - 23 T^{2} + 14 T^{3} + 2002 T^{4} + 14 p T^{5} - 23 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 41 T^{2} + 2964 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 15 T + 145 T^{2} - 1050 T^{3} + 6216 T^{4} - 1050 p T^{5} + 145 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 9 T + 21 T^{2} - 54 T^{3} - 1342 T^{4} - 54 p T^{5} + 21 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 3 T + 62 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 74 T^{2} + 112 T^{3} + 1327 T^{4} + 112 p T^{5} - 74 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 153 T^{2} + 14780 T^{4} - 153 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12 T + 130 T^{2} - 984 T^{3} + 4899 T^{4} - 984 p T^{5} + 130 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 15 T + 247 T^{2} + 2580 T^{3} + 29268 T^{4} + 2580 p T^{5} + 247 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 13 T + 194 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T + 174 T^{2} - 972 T^{3} + 19391 T^{4} - 972 p T^{5} + 174 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 173 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.007019867811029442905881992672, −7.943979476609237453463135496983, −7.72733367602777483952212944839, −7.55766195598256490420651133990, −7.30517058678654557493665942755, −7.11891868940192001033898936570, −6.66203133104305873301886822119, −6.42102894714823006412216017325, −6.34907847785201499892056574163, −5.75226885464158588717406942495, −5.73691770486390681324115885842, −4.77353454852939250129448873877, −4.71664751438050718649166275680, −4.48261076331967985419411705959, −4.06063035414967822877233216048, −3.98383083553465758531094576365, −3.84861126650066812172850259643, −3.41099309677987889613269482234, −2.79682407681572668185502401679, −2.76535669780637322480292955984, −2.57776381043428278930367253039, −2.28103475595676712536872874421, −1.70333434113284914696913424372, −1.38136662430918701728729888055, −0.15111995179643117920348690961,
0.15111995179643117920348690961, 1.38136662430918701728729888055, 1.70333434113284914696913424372, 2.28103475595676712536872874421, 2.57776381043428278930367253039, 2.76535669780637322480292955984, 2.79682407681572668185502401679, 3.41099309677987889613269482234, 3.84861126650066812172850259643, 3.98383083553465758531094576365, 4.06063035414967822877233216048, 4.48261076331967985419411705959, 4.71664751438050718649166275680, 4.77353454852939250129448873877, 5.73691770486390681324115885842, 5.75226885464158588717406942495, 6.34907847785201499892056574163, 6.42102894714823006412216017325, 6.66203133104305873301886822119, 7.11891868940192001033898936570, 7.30517058678654557493665942755, 7.55766195598256490420651133990, 7.72733367602777483952212944839, 7.943979476609237453463135496983, 8.007019867811029442905881992672
