| L(s) = 1 | + 3-s − 2·4-s − 5-s + 7-s + 9-s − 11-s − 2·12-s − 4·13-s − 15-s + 4·16-s + 3·17-s − 19-s + 2·20-s + 21-s − 3·23-s + 25-s + 27-s − 2·28-s − 9·29-s − 10·31-s − 33-s − 35-s − 2·36-s − 4·37-s − 4·39-s + 6·41-s − 43-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 1.10·13-s − 0.258·15-s + 16-s + 0.727·17-s − 0.229·19-s + 0.447·20-s + 0.218·21-s − 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 1.67·29-s − 1.79·31-s − 0.174·33-s − 0.169·35-s − 1/3·36-s − 0.657·37-s − 0.640·39-s + 0.937·41-s − 0.152·43-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−19.41471105467032, −18.88893778776954, −18.25462868702589, −17.58824084229461, −16.79578790948501, −16.14841419799811, −15.04144387258011, −14.60964811447990, −14.15560197263640, −13.09575940578327, −12.69115556600672, −11.85493503861050, −10.86202135415100, −9.971935831964999, −9.349195453153449, −8.596981698368817, −7.706221457938869, −7.363551265998250, −5.753208163254029, −4.973162294514953, −4.077913532393325, −3.247823276201408, −1.830559280227448, 0,
1.830559280227448, 3.247823276201408, 4.077913532393325, 4.973162294514953, 5.753208163254029, 7.363551265998250, 7.706221457938869, 8.596981698368817, 9.349195453153449, 9.971935831964999, 10.86202135415100, 11.85493503861050, 12.69115556600672, 13.09575940578327, 14.15560197263640, 14.60964811447990, 15.04144387258011, 16.14841419799811, 16.79578790948501, 17.58824084229461, 18.25462868702589, 18.88893778776954, 19.41471105467032
