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
−8.192032851112216973830186051376, −7.54054361859304036607750755648, −6.66951474604403914259980265722, −5.78657006035942566046847608494, −4.89236832560684339986415106799, −4.04228238736635998473333209074, −3.48843370538725787568465334940, −2.64600065094863566637987939235, −1.92117085377281792003959313709, 0,
1.92117085377281792003959313709, 2.64600065094863566637987939235, 3.48843370538725787568465334940, 4.04228238736635998473333209074, 4.89236832560684339986415106799, 5.78657006035942566046847608494, 6.66951474604403914259980265722, 7.54054361859304036607750755648, 8.192032851112216973830186051376