| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 2·5-s − 2·6-s + 9-s + 4·10-s − 5·11-s − 2·12-s + 13-s − 2·15-s − 4·16-s + 17-s + 2·18-s + 8·19-s + 4·20-s − 10·22-s + 6·23-s − 25-s + 2·26-s − 27-s − 8·29-s − 4·30-s + 9·31-s − 8·32-s + 5·33-s + 2·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.894·5-s − 0.816·6-s + 1/3·9-s + 1.26·10-s − 1.50·11-s − 0.577·12-s + 0.277·13-s − 0.516·15-s − 16-s + 0.242·17-s + 0.471·18-s + 1.83·19-s + 0.894·20-s − 2.13·22-s + 1.25·23-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 1.48·29-s − 0.730·30-s + 1.61·31-s − 1.41·32-s + 0.870·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.601849550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.601849550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.93016515323235, −14.49044678901256, −13.70633996945648, −13.37388442288513, −13.16432091685824, −12.61590946280538, −11.94866007442546, −11.35168071602105, −11.10865335330858, −10.29642343666915, −9.680957220135326, −9.435931421319286, −8.483984336099015, −7.652564915664916, −7.306195443100695, −6.416909948454265, −5.962202851789690, −5.424630373400217, −5.136864041002024, −4.561461894664600, −3.718464043902038, −2.910311745884203, −2.636424927601588, −1.610393376708326, −0.6636062013137042,
0.6636062013137042, 1.610393376708326, 2.636424927601588, 2.910311745884203, 3.718464043902038, 4.561461894664600, 5.136864041002024, 5.424630373400217, 5.962202851789690, 6.416909948454265, 7.306195443100695, 7.652564915664916, 8.483984336099015, 9.435931421319286, 9.680957220135326, 10.29642343666915, 11.10865335330858, 11.35168071602105, 11.94866007442546, 12.61590946280538, 13.16432091685824, 13.37388442288513, 13.70633996945648, 14.49044678901256, 14.93016515323235
