L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s + 8-s − 2·9-s − 3·10-s + 4·11-s − 12-s + 2·13-s + 3·15-s + 16-s − 7·17-s − 2·18-s + 4·19-s − 3·20-s + 4·22-s − 4·23-s − 24-s + 4·25-s + 2·26-s + 5·27-s + 9·29-s + 3·30-s + 31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.948·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.774·15-s + 1/4·16-s − 1.69·17-s − 0.471·18-s + 0.917·19-s − 0.670·20-s + 0.852·22-s − 0.834·23-s − 0.204·24-s + 4/5·25-s + 0.392·26-s + 0.962·27-s + 1.67·29-s + 0.547·30-s + 0.179·31-s + 0.176·32-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 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.077677168247136472411488625880, −7.17906424295495966564164952740, −6.33929390883062526296341267286, −6.11422338306747993709484741313, −4.74058507747248463442445289910, −4.44237954271210154053290210162, −3.54917187090303065610735328827, −2.80319377575133295406951816092, −1.33184807313600885384064697333, 0,
1.33184807313600885384064697333, 2.80319377575133295406951816092, 3.54917187090303065610735328827, 4.44237954271210154053290210162, 4.74058507747248463442445289910, 6.11422338306747993709484741313, 6.33929390883062526296341267286, 7.17906424295495966564164952740, 8.077677168247136472411488625880