| L(s) = 1 | − 2·7-s + 4·11-s − 6·13-s − 6·17-s + 4·23-s − 5·25-s + 4·29-s − 10·31-s − 2·37-s + 2·41-s + 8·43-s − 12·47-s − 3·49-s − 12·53-s + 4·59-s − 2·61-s + 4·67-s − 4·71-s − 10·73-s − 8·77-s + 6·79-s − 12·83-s − 2·89-s + 12·91-s − 6·97-s + 4·101-s + 10·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.834·23-s − 25-s + 0.742·29-s − 1.79·31-s − 0.328·37-s + 0.312·41-s + 1.21·43-s − 1.75·47-s − 3/7·49-s − 1.64·53-s + 0.520·59-s − 0.256·61-s + 0.488·67-s − 0.474·71-s − 1.17·73-s − 0.911·77-s + 0.675·79-s − 1.31·83-s − 0.211·89-s + 1.25·91-s − 0.609·97-s + 0.398·101-s + 0.985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.392805556443339877432708764972, −8.825925610400178898897528083054, −7.56237844869816331895572212869, −6.86617713649335228477722027286, −6.19174829158795733764599049329, −4.99207876302239290655432150674, −4.14300745369301184814326433789, −3.04605515966584879794132221618, −1.89290593869789230240007927288, 0,
1.89290593869789230240007927288, 3.04605515966584879794132221618, 4.14300745369301184814326433789, 4.99207876302239290655432150674, 6.19174829158795733764599049329, 6.86617713649335228477722027286, 7.56237844869816331895572212869, 8.825925610400178898897528083054, 9.392805556443339877432708764972
