| 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
−8.179782852143465266504137726561, −7.18865152305170992441539021464, −6.70575328522764976753560376616, −5.73338098252798190386130466263, −5.17931056662909807775657936859, −4.35769142890331167144865406178, −3.60220754430447676840995541021, −2.09240473980601433161203000166, −0.989229653631021658620912667436, 0,
0.989229653631021658620912667436, 2.09240473980601433161203000166, 3.60220754430447676840995541021, 4.35769142890331167144865406178, 5.17931056662909807775657936859, 5.73338098252798190386130466263, 6.70575328522764976753560376616, 7.18865152305170992441539021464, 8.179782852143465266504137726561
