| L(s) = 1 | + 2-s − 2·3-s + 4-s + 2·5-s − 2·6-s + 8-s + 9-s + 2·10-s − 4·11-s − 2·12-s − 13-s − 4·15-s + 16-s + 18-s + 6·19-s + 2·20-s − 4·22-s − 2·23-s − 2·24-s − 25-s − 26-s + 4·27-s − 5·29-s − 4·30-s − 4·31-s + 32-s + 8·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.577·12-s − 0.277·13-s − 1.03·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s + 0.447·20-s − 0.852·22-s − 0.417·23-s − 0.408·24-s − 1/5·25-s − 0.196·26-s + 0.769·27-s − 0.928·29-s − 0.730·30-s − 0.718·31-s + 0.176·32-s + 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70622142184209217312166391239, −7.27795394640900982572674914145, −6.20471154498088771771764694638, −5.73481153956338108677420667915, −5.27366859119609692845186449682, −4.63430201503440701455249800067, −3.40425258413077711487060916821, −2.52193527935593314887261757718, −1.50824860064404698122940785339, 0,
1.50824860064404698122940785339, 2.52193527935593314887261757718, 3.40425258413077711487060916821, 4.63430201503440701455249800067, 5.27366859119609692845186449682, 5.73481153956338108677420667915, 6.20471154498088771771764694638, 7.27795394640900982572674914145, 7.70622142184209217312166391239
