| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 2·11-s − 4·13-s − 15-s + 6·17-s + 6·19-s − 21-s + 8·23-s + 25-s − 27-s + 2·29-s − 10·31-s − 2·33-s + 35-s − 2·37-s + 4·39-s + 10·41-s − 4·43-s + 45-s + 8·47-s + 49-s − 6·51-s − 4·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.258·15-s + 1.45·17-s + 1.37·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.79·31-s − 0.348·33-s + 0.169·35-s − 0.328·37-s + 0.640·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.549·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.168594496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.168594496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.57565295991927829212361341289, −7.44892493942610284474973988155, −6.61501305905233192387918282561, −5.61584663845844552284711308292, −5.31433908581209118259282555585, −4.60231981508891573060399276015, −3.54154931087579933872337628887, −2.78338883078306886646704009036, −1.60416043078245411626546805768, −0.842665737921956982539486674155,
0.842665737921956982539486674155, 1.60416043078245411626546805768, 2.78338883078306886646704009036, 3.54154931087579933872337628887, 4.60231981508891573060399276015, 5.31433908581209118259282555585, 5.61584663845844552284711308292, 6.61501305905233192387918282561, 7.44892493942610284474973988155, 7.57565295991927829212361341289
