| L(s) = 1 | + 3·3-s − 7·7-s + 9·9-s − 26·11-s + 39·13-s + 101·17-s − 48·19-s − 21·21-s − 57·23-s + 27·27-s − 291·29-s − 79·31-s − 78·33-s − 322·37-s + 117·39-s + 455·41-s + 307·43-s − 508·47-s + 49·49-s + 303·51-s + 31·53-s − 144·57-s + 211·59-s − 797·61-s − 63·63-s + 924·67-s − 171·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.712·11-s + 0.832·13-s + 1.44·17-s − 0.579·19-s − 0.218·21-s − 0.516·23-s + 0.192·27-s − 1.86·29-s − 0.457·31-s − 0.411·33-s − 1.43·37-s + 0.480·39-s + 1.73·41-s + 1.08·43-s − 1.57·47-s + 1/7·49-s + 0.831·51-s + 0.0803·53-s − 0.334·57-s + 0.465·59-s − 1.67·61-s − 0.125·63-s + 1.68·67-s − 0.298·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 101 T + p^{3} T^{2} \) |
| 19 | \( 1 + 48 T + p^{3} T^{2} \) |
| 23 | \( 1 + 57 T + p^{3} T^{2} \) |
| 29 | \( 1 + 291 T + p^{3} T^{2} \) |
| 31 | \( 1 + 79 T + p^{3} T^{2} \) |
| 37 | \( 1 + 322 T + p^{3} T^{2} \) |
| 41 | \( 1 - 455 T + p^{3} T^{2} \) |
| 43 | \( 1 - 307 T + p^{3} T^{2} \) |
| 47 | \( 1 + 508 T + p^{3} T^{2} \) |
| 53 | \( 1 - 31 T + p^{3} T^{2} \) |
| 59 | \( 1 - 211 T + p^{3} T^{2} \) |
| 61 | \( 1 + 797 T + p^{3} T^{2} \) |
| 67 | \( 1 - 924 T + p^{3} T^{2} \) |
| 71 | \( 1 - 212 T + p^{3} T^{2} \) |
| 73 | \( 1 + 22 T + p^{3} T^{2} \) |
| 79 | \( 1 - 874 T + p^{3} T^{2} \) |
| 83 | \( 1 - 587 T + p^{3} T^{2} \) |
| 89 | \( 1 + 266 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1822 T + p^{3} T^{2} \) |
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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
−8.204223376201054857905827035118, −7.78789721677325686217128866770, −6.90155183786898252037215267280, −5.90096837887017076649224271666, −5.29651482453059819919069255760, −3.98683928744130890965534548839, −3.43725513853501899033649223810, −2.40556545567456754515392357491, −1.36628170137343436713209295371, 0,
1.36628170137343436713209295371, 2.40556545567456754515392357491, 3.43725513853501899033649223810, 3.98683928744130890965534548839, 5.29651482453059819919069255760, 5.90096837887017076649224271666, 6.90155183786898252037215267280, 7.78789721677325686217128866770, 8.204223376201054857905827035118
