| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 7-s + 8-s − 2·9-s − 3·10-s + 12-s − 13-s + 14-s − 3·15-s + 16-s + 3·17-s − 2·18-s − 2·19-s − 3·20-s + 21-s + 24-s + 4·25-s − 26-s − 5·27-s + 28-s − 6·29-s − 3·30-s − 4·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 0.458·19-s − 0.670·20-s + 0.218·21-s + 0.204·24-s + 4/5·25-s − 0.196·26-s − 0.962·27-s + 0.188·28-s − 1.11·29-s − 0.547·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3146 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3146 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.158308298923332532258561346975, −7.61357841654986180404055851494, −6.99995607550208376322691762678, −5.85883048660862822422904238763, −5.16078862035801732240851587336, −4.20840140522021551558258294497, −3.58506615738380785640972516083, −2.89028538035854279081544483316, −1.75251405579569256697321594310, 0,
1.75251405579569256697321594310, 2.89028538035854279081544483316, 3.58506615738380785640972516083, 4.20840140522021551558258294497, 5.16078862035801732240851587336, 5.85883048660862822422904238763, 6.99995607550208376322691762678, 7.61357841654986180404055851494, 8.158308298923332532258561346975
