L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 3·8-s − 2·9-s − 10-s − 6·11-s − 12-s + 13-s − 15-s − 16-s − 17-s − 2·18-s − 2·19-s + 20-s − 6·22-s + 2·23-s − 3·24-s − 4·25-s + 26-s − 5·27-s + 6·29-s − 30-s + 2·31-s + 5·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.06·8-s − 2/3·9-s − 0.316·10-s − 1.80·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s − 1/4·16-s − 0.242·17-s − 0.471·18-s − 0.458·19-s + 0.223·20-s − 1.27·22-s + 0.417·23-s − 0.612·24-s − 4/5·25-s + 0.196·26-s − 0.962·27-s + 1.11·29-s − 0.182·30-s + 0.359·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.984357725905159140193605100133, −8.846842789582778602892056433896, −8.340043193019847775924528656408, −7.55055483522611773628657287970, −6.14951158357540413540768917282, −5.29636676705374581558916997454, −4.43250716134528419079552745204, −3.31538254167605948583204104793, −2.54907770554245923576810875576, 0,
2.54907770554245923576810875576, 3.31538254167605948583204104793, 4.43250716134528419079552745204, 5.29636676705374581558916997454, 6.14951158357540413540768917282, 7.55055483522611773628657287970, 8.340043193019847775924528656408, 8.846842789582778602892056433896, 9.984357725905159140193605100133