| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 7-s + 9-s − 2·10-s + 11-s + 2·12-s − 6·13-s − 2·14-s + 15-s − 4·16-s − 7·17-s − 2·18-s − 5·19-s + 2·20-s + 21-s − 2·22-s − 23-s + 25-s + 12·26-s + 27-s + 2·28-s − 5·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s − 1.66·13-s − 0.534·14-s + 0.258·15-s − 16-s − 1.69·17-s − 0.471·18-s − 1.14·19-s + 0.447·20-s + 0.218·21-s − 0.426·22-s − 0.208·23-s + 1/5·25-s + 2.35·26-s + 0.192·27-s + 0.377·28-s − 0.928·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−19.47944521981931, −18.84118986011334, −18.14382745077329, −17.40267506773015, −17.15217674499403, −16.33971671531793, −15.40680978306617, −14.75176607943080, −14.10634164600144, −13.14775504443514, −12.52650706133239, −11.31069054862254, −10.77044086529249, −9.925305511860557, −9.212018498778742, −8.846741941824141, −7.870556628042501, −7.221117898961584, −6.444950830832669, −5.000306352651380, −4.127907888701044, −2.369597497356915, −1.874727272896544, 0,
1.874727272896544, 2.369597497356915, 4.127907888701044, 5.000306352651380, 6.444950830832669, 7.221117898961584, 7.870556628042501, 8.846741941824141, 9.212018498778742, 9.925305511860557, 10.77044086529249, 11.31069054862254, 12.52650706133239, 13.14775504443514, 14.10634164600144, 14.75176607943080, 15.40680978306617, 16.33971671531793, 17.15217674499403, 17.40267506773015, 18.14382745077329, 18.84118986011334, 19.47944521981931
