L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 2·13-s + 16-s − 2·19-s − 20-s + 25-s − 2·26-s − 6·29-s − 8·31-s + 32-s − 4·37-s − 2·38-s − 40-s + 6·41-s + 2·43-s − 6·47-s + 50-s − 2·52-s − 6·53-s − 6·58-s + 12·59-s − 8·61-s − 8·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.554·13-s + 1/4·16-s − 0.458·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.657·37-s − 0.324·38-s − 0.158·40-s + 0.937·41-s + 0.304·43-s − 0.875·47-s + 0.141·50-s − 0.277·52-s − 0.824·53-s − 0.787·58-s + 1.56·59-s − 1.02·61-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−7.72884256525441388224603340660, −7.31197475215030351263291600034, −6.49895271238158734940980188459, −5.67215196547072475539085458023, −5.00679853986208418603505167842, −4.16364492087773222905282744018, −3.52571722299951763856854309445, −2.57091189800820073190585026882, −1.61549144219507342431900476742, 0,
1.61549144219507342431900476742, 2.57091189800820073190585026882, 3.52571722299951763856854309445, 4.16364492087773222905282744018, 5.00679853986208418603505167842, 5.67215196547072475539085458023, 6.49895271238158734940980188459, 7.31197475215030351263291600034, 7.72884256525441388224603340660