L(s) = 1 | − 2-s − 2·3-s + 4-s + 5-s + 2·6-s − 2·7-s − 8-s + 9-s − 10-s − 2·11-s − 2·12-s − 6·13-s + 2·14-s − 2·15-s + 16-s + 17-s − 18-s − 8·19-s + 20-s + 4·21-s + 2·22-s − 2·23-s + 2·24-s + 25-s + 6·26-s + 4·27-s − 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.577·12-s − 1.66·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.83·19-s + 0.223·20-s + 0.872·21-s + 0.426·22-s − 0.417·23-s + 0.408·24-s + 1/5·25-s + 1.17·26-s + 0.769·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.31532200871767137866028067647, −11.07165632295697513016898928455, −10.26417859843699330979855735854, −9.549933585998417375570109355541, −8.148249676902755756251024022743, −6.81413271719296947895724716147, −6.03307727148352233442601430726, −4.81574131972665003276755918497, −2.53723446260811662839626731568, 0,
2.53723446260811662839626731568, 4.81574131972665003276755918497, 6.03307727148352233442601430726, 6.81413271719296947895724716147, 8.148249676902755756251024022743, 9.549933585998417375570109355541, 10.26417859843699330979855735854, 11.07165632295697513016898928455, 12.31532200871767137866028067647