| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s + 8·7-s − 2·8-s + 3·9-s + 2·12-s − 12·13-s + 16·14-s − 4·16-s − 12·17-s + 6·18-s − 2·19-s + 16·21-s − 4·24-s − 8·25-s − 24·26-s + 10·27-s + 8·28-s − 2·32-s − 24·34-s + 3·36-s − 4·38-s − 24·39-s − 6·41-s + 32·42-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s + 3.02·7-s − 0.707·8-s + 9-s + 0.577·12-s − 3.32·13-s + 4.27·14-s − 16-s − 2.91·17-s + 1.41·18-s − 0.458·19-s + 3.49·21-s − 0.816·24-s − 8/5·25-s − 4.70·26-s + 1.92·27-s + 1.51·28-s − 0.353·32-s − 4.11·34-s + 1/2·36-s − 0.648·38-s − 3.84·39-s − 0.937·41-s + 4.93·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.003342403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.003342403\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^3$ | \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 507 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 86 T^{2} + 456 T^{3} + 1971 T^{4} + 456 p T^{5} + 86 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 4 T^{2} - 513 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3810 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T - 14 T^{2} + 64 T^{3} + 1483 T^{4} + 64 p T^{5} - 14 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 56 T^{2} - 96 T^{3} + 111 T^{4} - 96 p T^{5} + 56 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 26 T^{2} + 144 T^{3} + 3483 T^{4} + 144 p T^{5} + 26 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 18 T + 131 T^{2} - 1350 T^{3} + 14652 T^{4} - 1350 p T^{5} + 131 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 68 T^{2} - 80 T^{3} + 9799 T^{4} - 80 p T^{5} - 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 714 T^{3} + 10476 T^{4} + 714 p T^{5} + 131 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 2 T - 47 T^{2} + 190 T^{3} - 3020 T^{4} + 190 p T^{5} - 47 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 + 86 T^{2} + 1155 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 278 T^{2} - 2880 T^{3} + 29619 T^{4} - 2880 p T^{5} + 278 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 18 T + 257 T^{2} + 2682 T^{3} + 23268 T^{4} + 2682 p T^{5} + 257 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04563987942332538864544266026, −9.594401128101041311197933591779, −9.443051614271687020926982601934, −9.206538024876112526840697440671, −8.886082651476942872306757100776, −8.519337754733934367291991047061, −8.119244971647420069877070792288, −8.038604944968059247643151268766, −7.975051088910496573396710006444, −7.37587048204865065347075386722, −7.00212381133974054533424786471, −6.93738025068783551799881966503, −6.62789165325942128680394582565, −5.91312693089997754829289866918, −5.57999378999719840774515171077, −5.05717147512744661904457551374, −4.88440965723369080757589943641, −4.74228246569414955471926197973, −4.44535338726630202289195031464, −4.14371224039697263635162358299, −3.82709755954716876771361966832, −2.95364971832215458448376476736, −2.32452839037930250275437655529, −2.23429147108818505649651132313, −1.97480694420883125076863906035,
1.97480694420883125076863906035, 2.23429147108818505649651132313, 2.32452839037930250275437655529, 2.95364971832215458448376476736, 3.82709755954716876771361966832, 4.14371224039697263635162358299, 4.44535338726630202289195031464, 4.74228246569414955471926197973, 4.88440965723369080757589943641, 5.05717147512744661904457551374, 5.57999378999719840774515171077, 5.91312693089997754829289866918, 6.62789165325942128680394582565, 6.93738025068783551799881966503, 7.00212381133974054533424786471, 7.37587048204865065347075386722, 7.975051088910496573396710006444, 8.038604944968059247643151268766, 8.119244971647420069877070792288, 8.519337754733934367291991047061, 8.886082651476942872306757100776, 9.206538024876112526840697440671, 9.443051614271687020926982601934, 9.594401128101041311197933591779, 10.04563987942332538864544266026
