L(s) = 1 | − 2-s − 3·5-s + 2·7-s + 8-s + 3·10-s + 12·11-s − 5·13-s − 2·14-s − 16-s − 6·17-s − 19-s − 12·22-s − 3·23-s + 5·25-s + 5·26-s + 6·29-s − 8·31-s + 6·34-s − 6·35-s + 16·37-s + 38-s − 3·40-s + 6·41-s + 4·43-s + 3·46-s + 6·47-s + 3·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.34·5-s + 0.755·7-s + 0.353·8-s + 0.948·10-s + 3.61·11-s − 1.38·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s − 0.229·19-s − 2.55·22-s − 0.625·23-s + 25-s + 0.980·26-s + 1.11·29-s − 1.43·31-s + 1.02·34-s − 1.01·35-s + 2.63·37-s + 0.162·38-s − 0.474·40-s + 0.937·41-s + 0.609·43-s + 0.442·46-s + 0.875·47-s + 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.136494954\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.136494954\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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
−9.065089087978926621678223299756, −8.877290747695455478508886466250, −8.478964609417662127624443359851, −8.088845259334253567837359767743, −7.59466234094809917145286280806, −7.17115200286137465120695092347, −7.14586737733098363587534372286, −6.53978376501671198090063981037, −6.26617674338902892488988018934, −5.74560766387340975041462990873, −5.12291520735608662543774538075, −4.32033307344482117162599873241, −4.30678440294243884799756986519, −4.08514119385663899794742989753, −3.83647788796639214552358228690, −2.64430404921242313233186325293, −2.56310454661527329649042056031, −1.49794307760171434164442726551, −1.29042999942740949366716676356, −0.45862839633879342325185375475,
0.45862839633879342325185375475, 1.29042999942740949366716676356, 1.49794307760171434164442726551, 2.56310454661527329649042056031, 2.64430404921242313233186325293, 3.83647788796639214552358228690, 4.08514119385663899794742989753, 4.30678440294243884799756986519, 4.32033307344482117162599873241, 5.12291520735608662543774538075, 5.74560766387340975041462990873, 6.26617674338902892488988018934, 6.53978376501671198090063981037, 7.14586737733098363587534372286, 7.17115200286137465120695092347, 7.59466234094809917145286280806, 8.088845259334253567837359767743, 8.478964609417662127624443359851, 8.877290747695455478508886466250, 9.065089087978926621678223299756