| L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s − 6·13-s + 6·17-s + 2·21-s − 4·23-s − 5·25-s − 27-s − 4·29-s − 10·31-s + 4·33-s − 2·37-s + 6·39-s − 2·41-s + 8·43-s + 12·47-s − 3·49-s − 6·51-s + 12·53-s − 4·59-s − 2·61-s − 2·63-s + 4·67-s + 4·69-s + 4·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.436·21-s − 0.834·23-s − 25-s − 0.192·27-s − 0.742·29-s − 1.79·31-s + 0.696·33-s − 0.328·37-s + 0.960·39-s − 0.312·41-s + 1.21·43-s + 1.75·47-s − 3/7·49-s − 0.840·51-s + 1.64·53-s − 0.520·59-s − 0.256·61-s − 0.251·63-s + 0.488·67-s + 0.481·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 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
−10.72157176858504266602545238224, −10.03012842475637444340101123739, −9.355375973689939441990781827382, −7.73185283482156320096864018539, −7.29744951786402483785793531451, −5.81271121311664380914911812156, −5.25073137830748218691985779847, −3.77490371552045331982722842815, −2.36547685273488160999702672858, 0,
2.36547685273488160999702672858, 3.77490371552045331982722842815, 5.25073137830748218691985779847, 5.81271121311664380914911812156, 7.29744951786402483785793531451, 7.73185283482156320096864018539, 9.355375973689939441990781827382, 10.03012842475637444340101123739, 10.72157176858504266602545238224
