| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s − 7·13-s + 15-s − 17-s − 3·19-s − 21-s + 9·23-s − 4·25-s + 27-s + 6·29-s − 4·31-s + 33-s − 35-s − 10·37-s − 7·39-s + 3·41-s − 3·43-s + 45-s + 8·47-s + 49-s − 51-s − 4·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.94·13-s + 0.258·15-s − 0.242·17-s − 0.688·19-s − 0.218·21-s + 1.87·23-s − 4/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.174·33-s − 0.169·35-s − 1.64·37-s − 1.12·39-s + 0.468·41-s − 0.457·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.140·51-s − 0.549·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.68947041499449691960054903701, −7.02935945178998292751328337282, −6.58776159278713151014537525243, −5.50273371009665926975748123563, −4.85984761454734213808896082899, −4.09164158640390968298524855507, −3.02222688047841876990941790657, −2.48060593933830004359465467484, −1.52171354161658990095629856102, 0,
1.52171354161658990095629856102, 2.48060593933830004359465467484, 3.02222688047841876990941790657, 4.09164158640390968298524855507, 4.85984761454734213808896082899, 5.50273371009665926975748123563, 6.58776159278713151014537525243, 7.02935945178998292751328337282, 7.68947041499449691960054903701
