| L(s) = 1 | + 2·3-s − 7-s + 9-s − 11-s − 4·13-s − 6·19-s − 2·21-s + 3·23-s − 4·27-s − 3·29-s − 2·33-s − 9·37-s − 8·39-s + 2·41-s + 9·43-s − 6·47-s + 49-s − 6·53-s − 12·57-s − 8·59-s − 10·61-s − 63-s − 67-s + 6·69-s + 7·71-s + 2·73-s + 77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 1.37·19-s − 0.436·21-s + 0.625·23-s − 0.769·27-s − 0.557·29-s − 0.348·33-s − 1.47·37-s − 1.28·39-s + 0.312·41-s + 1.37·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.58·57-s − 1.04·59-s − 1.28·61-s − 0.125·63-s − 0.122·67-s + 0.722·69-s + 0.830·71-s + 0.234·73-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.425964942761596266734775731379, −7.77858113420674514527930321373, −7.08810586050844792675815557685, −6.23501293850229139396092656147, −5.22657965740984814079675950716, −4.35085225209188815409729106643, −3.40524423690805139640810994027, −2.65429870580906953006120844335, −1.88455274580094335413583729871, 0,
1.88455274580094335413583729871, 2.65429870580906953006120844335, 3.40524423690805139640810994027, 4.35085225209188815409729106643, 5.22657965740984814079675950716, 6.23501293850229139396092656147, 7.08810586050844792675815557685, 7.77858113420674514527930321373, 8.425964942761596266734775731379
