| L(s) = 1 | + 4·11-s − 13-s + 2·17-s + 4·19-s − 8·23-s + 2·29-s + 8·31-s − 6·37-s + 6·41-s − 4·43-s + 8·47-s − 7·49-s + 6·53-s − 12·59-s − 2·61-s − 4·67-s + 6·73-s − 16·79-s + 4·83-s − 10·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s − 49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s − 0.488·67-s + 0.702·73-s − 1.80·79-s + 0.439·83-s − 1.05·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.80475473906593, −14.20184391008593, −13.87503386846397, −13.59261774836330, −12.60399182105150, −12.13404957507162, −11.92995070630233, −11.38576697068055, −10.63244235777004, −10.11549145228838, −9.625187264866806, −9.225871875736002, −8.471590530963014, −8.032190719476675, −7.438932088985554, −6.811579395709945, −6.293892497945542, −5.735695281840401, −5.136252256205316, −4.286965630775291, −3.996185936114503, −3.155856807892431, −2.585824994444684, −1.607810911826166, −1.112086997296561, 0,
1.112086997296561, 1.607810911826166, 2.585824994444684, 3.155856807892431, 3.996185936114503, 4.286965630775291, 5.136252256205316, 5.735695281840401, 6.293892497945542, 6.811579395709945, 7.438932088985554, 8.032190719476675, 8.471590530963014, 9.225871875736002, 9.625187264866806, 10.11549145228838, 10.63244235777004, 11.38576697068055, 11.92995070630233, 12.13404957507162, 12.60399182105150, 13.59261774836330, 13.87503386846397, 14.20184391008593, 14.80475473906593
