| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s + 3·7-s + 3·8-s + 9-s − 10-s − 12-s − 3·14-s + 15-s − 16-s + 6·17-s − 18-s − 3·19-s − 20-s + 3·21-s + 3·24-s − 3·25-s + 4·27-s − 3·28-s − 30-s − 5·32-s − 6·34-s + 3·35-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.801·14-s + 0.258·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.688·19-s − 0.223·20-s + 0.654·21-s + 0.612·24-s − 3/5·25-s + 0.769·27-s − 0.566·28-s − 0.182·30-s − 0.883·32-s − 1.02·34-s + 0.507·35-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.635228035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635228035\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 53 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 96 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 123 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 84 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.043353359391725669426856710212, −8.641723828824264234136409871105, −8.330770535373514820161957999349, −7.87833615583389522336991059166, −7.40065935312884365907935606812, −7.08722274830578162628258625454, −6.15318864421701734951420204618, −5.65976074603391313378814069361, −5.12148369184527315335193795403, −4.56475499232724808268733222437, −4.05208077345204490051413133857, −3.39520893518107047723812797861, −2.45314448912986295003954781704, −1.75423769764077994727717318266, −1.00715275245516674308016097155,
1.00715275245516674308016097155, 1.75423769764077994727717318266, 2.45314448912986295003954781704, 3.39520893518107047723812797861, 4.05208077345204490051413133857, 4.56475499232724808268733222437, 5.12148369184527315335193795403, 5.65976074603391313378814069361, 6.15318864421701734951420204618, 7.08722274830578162628258625454, 7.40065935312884365907935606812, 7.87833615583389522336991059166, 8.330770535373514820161957999349, 8.641723828824264234136409871105, 9.043353359391725669426856710212
