L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·5-s − 2·6-s + 5·7-s + 4·8-s − 2·9-s − 4·10-s + 2·11-s − 3·12-s + 9·13-s + 10·14-s + 2·15-s + 5·16-s − 11·17-s − 4·18-s + 2·19-s − 6·20-s − 5·21-s + 4·22-s − 5·23-s − 4·24-s + 3·25-s + 18·26-s + 2·27-s + 15·28-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.894·5-s − 0.816·6-s + 1.88·7-s + 1.41·8-s − 2/3·9-s − 1.26·10-s + 0.603·11-s − 0.866·12-s + 2.49·13-s + 2.67·14-s + 0.516·15-s + 5/4·16-s − 2.66·17-s − 0.942·18-s + 0.458·19-s − 1.34·20-s − 1.09·21-s + 0.852·22-s − 1.04·23-s − 0.816·24-s + 3/5·25-s + 3.53·26-s + 0.384·27-s + 2.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64480900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64480900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.623907563\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.623907563\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 11 T + 61 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + p T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 51 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 118 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 143 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 145 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 169 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 221 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.73095394488533060482840740193, −7.60997155684186407723572968197, −7.38728154846066681298629759869, −6.89385089455797984941546809682, −6.23594943561140656061018942714, −6.20949244460525138115936996798, −5.84235918176609777813273539620, −5.83887233496711155911273628733, −4.96288219036884347543934071326, −4.85565670872468537898185768912, −4.41342022962806531521899613326, −4.06084226957362124677624646274, −3.99862851892922903040239489180, −3.64414265770998524740391324811, −2.86701457690197315479218648258, −2.58329835328814758685505668372, −1.99066043988537744338462474722, −1.73169345672928673189210497364, −0.937137535955367956210019968526, −0.71497397586316138786609257081,
0.71497397586316138786609257081, 0.937137535955367956210019968526, 1.73169345672928673189210497364, 1.99066043988537744338462474722, 2.58329835328814758685505668372, 2.86701457690197315479218648258, 3.64414265770998524740391324811, 3.99862851892922903040239489180, 4.06084226957362124677624646274, 4.41342022962806531521899613326, 4.85565670872468537898185768912, 4.96288219036884347543934071326, 5.83887233496711155911273628733, 5.84235918176609777813273539620, 6.20949244460525138115936996798, 6.23594943561140656061018942714, 6.89385089455797984941546809682, 7.38728154846066681298629759869, 7.60997155684186407723572968197, 7.73095394488533060482840740193