| L(s) = 1 | − 3-s − 2·7-s + 9-s − 6·11-s + 2·13-s + 6·17-s + 4·19-s + 2·21-s + 8·23-s − 27-s − 8·31-s + 6·33-s − 2·37-s − 2·39-s − 6·41-s − 4·43-s + 4·47-s − 3·49-s − 6·51-s − 6·53-s − 4·57-s − 6·59-s − 6·61-s − 2·63-s − 8·69-s − 4·71-s − 12·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.436·21-s + 1.66·23-s − 0.192·27-s − 1.43·31-s + 1.04·33-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.781·59-s − 0.768·61-s − 0.251·63-s − 0.963·69-s − 0.474·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.581121196771271728980142550881, −7.58143893892209121895073378690, −7.20207685475442227431828586556, −6.08198532619333796070347285722, −5.41390091613278019584528777631, −4.89368686770547847465596752720, −3.41622971494981084233413281923, −2.95921669963291858838860415843, −1.37456716929394551516171235179, 0,
1.37456716929394551516171235179, 2.95921669963291858838860415843, 3.41622971494981084233413281923, 4.89368686770547847465596752720, 5.41390091613278019584528777631, 6.08198532619333796070347285722, 7.20207685475442227431828586556, 7.58143893892209121895073378690, 8.581121196771271728980142550881
