L(s) = 1 | + 2-s + 4-s + 4·5-s + 8-s − 3·9-s + 4·10-s − 2·11-s + 6·13-s + 16-s + 6·17-s − 3·18-s − 4·19-s + 4·20-s − 2·22-s + 8·23-s + 11·25-s + 6·26-s − 8·29-s + 32-s + 6·34-s − 3·36-s − 2·37-s − 4·38-s + 4·40-s + 41-s − 8·43-s − 2·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.353·8-s − 9-s + 1.26·10-s − 0.603·11-s + 1.66·13-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.917·19-s + 0.894·20-s − 0.426·22-s + 1.66·23-s + 11/5·25-s + 1.17·26-s − 1.48·29-s + 0.176·32-s + 1.02·34-s − 1/2·36-s − 0.328·37-s − 0.648·38-s + 0.632·40-s + 0.156·41-s − 1.21·43-s − 0.301·44-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
\(L(1)\) |
\(\approx\) |
\(4.433927792\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.433927792\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.613816602334229686054100646789, −7.65721528955671107769156120958, −6.56631387530868863503842917299, −6.11127408256228484107874823281, −5.40935598844077101472264620739, −5.12721343255384048228575572204, −3.64099149589075833458053763761, −2.99944773900680661378778345491, −2.09567390840630364723470427188, −1.18690792211090642717452721795,
1.18690792211090642717452721795, 2.09567390840630364723470427188, 2.99944773900680661378778345491, 3.64099149589075833458053763761, 5.12721343255384048228575572204, 5.40935598844077101472264620739, 6.11127408256228484107874823281, 6.56631387530868863503842917299, 7.65721528955671107769156120958, 8.613816602334229686054100646789