| L(s) = 1 | + 3·3-s + 12·5-s − 22·7-s + 9·9-s − 11·11-s − 48·13-s + 36·15-s − 54·17-s − 100·19-s − 66·21-s − 58·23-s + 19·25-s + 27·27-s + 262·29-s − 248·31-s − 33·33-s − 264·35-s − 130·37-s − 144·39-s − 26·41-s − 216·43-s + 108·45-s − 22·47-s + 141·49-s − 162·51-s + 620·53-s − 132·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.07·5-s − 1.18·7-s + 1/3·9-s − 0.301·11-s − 1.02·13-s + 0.619·15-s − 0.770·17-s − 1.20·19-s − 0.685·21-s − 0.525·23-s + 0.151·25-s + 0.192·27-s + 1.67·29-s − 1.43·31-s − 0.174·33-s − 1.27·35-s − 0.577·37-s − 0.591·39-s − 0.0990·41-s − 0.766·43-s + 0.357·45-s − 0.0682·47-s + 0.411·49-s − 0.444·51-s + 1.60·53-s − 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 13 | \( 1 + 48 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 262 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 130 T + p^{3} T^{2} \) |
| 41 | \( 1 + 26 T + p^{3} T^{2} \) |
| 43 | \( 1 + 216 T + p^{3} T^{2} \) |
| 47 | \( 1 + 22 T + p^{3} T^{2} \) |
| 53 | \( 1 - 620 T + p^{3} T^{2} \) |
| 59 | \( 1 - 424 T + p^{3} T^{2} \) |
| 61 | \( 1 - 340 T + p^{3} T^{2} \) |
| 67 | \( 1 - 620 T + p^{3} T^{2} \) |
| 71 | \( 1 + 810 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1118 T + p^{3} T^{2} \) |
| 79 | \( 1 - 214 T + p^{3} T^{2} \) |
| 83 | \( 1 + 988 T + p^{3} T^{2} \) |
| 89 | \( 1 + 6 T + p^{3} T^{2} \) |
| 97 | \( 1 - 590 T + p^{3} T^{2} \) |
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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
−10.06576069169886812431751195154, −9.185040791297553022850173063386, −8.433313675826933923844699277498, −7.08098258692572853694112298173, −6.43363866028503126802006253197, −5.38108779198416167704210257084, −4.11858733623833210058307842858, −2.79019634165152490258812906174, −2.01389133395057008493081491340, 0,
2.01389133395057008493081491340, 2.79019634165152490258812906174, 4.11858733623833210058307842858, 5.38108779198416167704210257084, 6.43363866028503126802006253197, 7.08098258692572853694112298173, 8.433313675826933923844699277498, 9.185040791297553022850173063386, 10.06576069169886812431751195154
