| L(s) = 1 | − 3-s + 9-s − 2·11-s − 4·13-s + 2·17-s − 2·19-s + 4·23-s − 27-s − 6·29-s − 2·31-s + 2·33-s + 10·37-s + 4·39-s + 10·41-s − 12·43-s + 8·47-s − 2·51-s + 2·57-s + 8·59-s − 2·61-s + 12·67-s − 4·69-s + 10·71-s + 4·73-s + 81-s − 12·83-s + 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.485·17-s − 0.458·19-s + 0.834·23-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.348·33-s + 1.64·37-s + 0.640·39-s + 1.56·41-s − 1.82·43-s + 1.16·47-s − 0.280·51-s + 0.264·57-s + 1.04·59-s − 0.256·61-s + 1.46·67-s − 0.481·69-s + 1.18·71-s + 0.468·73-s + 1/9·81-s − 1.31·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.88147028072409, −12.72163011199532, −12.36980610751627, −11.63377313853532, −11.23261559603780, −11.00178381407037, −10.28390325094814, −9.974885614824240, −9.427026914328777, −9.131499414306483, −8.302719497966910, −7.917710413689335, −7.433421941826638, −6.986553157221281, −6.525660526891353, −5.839110933940824, −5.375365594720447, −5.111687733918547, −4.404957075394291, −3.977661088998568, −3.297137374966092, −2.513557060655126, −2.299848758503706, −1.371013976323694, −0.6968457957030030, 0,
0.6968457957030030, 1.371013976323694, 2.299848758503706, 2.513557060655126, 3.297137374966092, 3.977661088998568, 4.404957075394291, 5.111687733918547, 5.375365594720447, 5.839110933940824, 6.525660526891353, 6.986553157221281, 7.433421941826638, 7.917710413689335, 8.302719497966910, 9.131499414306483, 9.427026914328777, 9.974885614824240, 10.28390325094814, 11.00178381407037, 11.23261559603780, 11.63377313853532, 12.36980610751627, 12.72163011199532, 12.88147028072409
