| L(s) = 1 | − 3-s − 2·7-s + 9-s + 13-s + 2·17-s + 2·19-s + 2·21-s − 8·23-s − 5·25-s − 27-s + 6·29-s − 2·31-s − 6·37-s − 39-s − 4·43-s − 8·47-s − 3·49-s − 2·51-s − 6·53-s − 2·57-s − 4·59-s + 2·61-s − 2·63-s − 2·67-s + 8·69-s − 4·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.485·17-s + 0.458·19-s + 0.436·21-s − 1.66·23-s − 25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.986·37-s − 0.160·39-s − 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s − 0.264·57-s − 0.520·59-s + 0.256·61-s − 0.251·63-s − 0.244·67-s + 0.963·69-s − 0.474·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.586196101860007060944293941872, −8.404773792929967833538138718608, −7.65338740605061265659475162578, −6.61277232194412223963156068467, −6.03523704391687496414629389438, −5.14953177527932686418168257619, −4.03288470481684680307144406184, −3.13736585970066578227810367428, −1.65921780267175436554426250858, 0,
1.65921780267175436554426250858, 3.13736585970066578227810367428, 4.03288470481684680307144406184, 5.14953177527932686418168257619, 6.03523704391687496414629389438, 6.61277232194412223963156068467, 7.65338740605061265659475162578, 8.404773792929967833538138718608, 9.586196101860007060944293941872
