| L(s) = 1 | − 2-s + 2·3-s − 4-s − 2·6-s + 3·8-s + 9-s − 2·11-s − 2·12-s − 16-s − 18-s + 6·19-s + 2·22-s − 6·23-s + 6·24-s − 4·27-s − 6·29-s + 6·31-s − 5·32-s − 4·33-s − 36-s − 6·37-s − 6·38-s − 8·41-s + 6·43-s + 2·44-s + 6·46-s + 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.816·6-s + 1.06·8-s + 1/3·9-s − 0.603·11-s − 0.577·12-s − 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.426·22-s − 1.25·23-s + 1.22·24-s − 0.769·27-s − 1.11·29-s + 1.07·31-s − 0.883·32-s − 0.696·33-s − 1/6·36-s − 0.986·37-s − 0.973·38-s − 1.24·41-s + 0.914·43-s + 0.301·44-s + 0.884·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.068191827134391210656761619105, −7.75231206828974756831516572422, −6.94193906820482250940695112154, −5.71189271864577058290855726039, −5.03938439449434719677819531447, −4.01835830666713836904412409867, −3.34846140832549633793259555127, −2.38777791858018364958111219742, −1.42856855966928046265001616841, 0,
1.42856855966928046265001616841, 2.38777791858018364958111219742, 3.34846140832549633793259555127, 4.01835830666713836904412409867, 5.03938439449434719677819531447, 5.71189271864577058290855726039, 6.94193906820482250940695112154, 7.75231206828974756831516572422, 8.068191827134391210656761619105
