| L(s) = 1 | + 2-s − 4-s + 4·7-s − 3·8-s − 3·9-s + 2·13-s + 4·14-s − 16-s − 3·18-s − 4·19-s + 4·23-s + 2·26-s − 4·28-s − 6·29-s − 4·31-s + 5·32-s + 3·36-s − 2·37-s − 4·38-s + 6·41-s − 4·43-s + 4·46-s + 9·49-s − 2·52-s − 6·53-s − 12·56-s − 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.51·7-s − 1.06·8-s − 9-s + 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.707·18-s − 0.917·19-s + 0.834·23-s + 0.392·26-s − 0.755·28-s − 1.11·29-s − 0.718·31-s + 0.883·32-s + 1/2·36-s − 0.328·37-s − 0.648·38-s + 0.937·41-s − 0.609·43-s + 0.589·46-s + 9/7·49-s − 0.277·52-s − 0.824·53-s − 1.60·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.78459739864314372018344946247, −6.71689283504453592886645528269, −5.87528865673177748562799631708, −5.37915785401627898631541565252, −4.76719005277138963416770734827, −4.07700453348399448284253011445, −3.32047709320597277659233545794, −2.38310664387109231415778319293, −1.37726342471784917548393521584, 0,
1.37726342471784917548393521584, 2.38310664387109231415778319293, 3.32047709320597277659233545794, 4.07700453348399448284253011445, 4.76719005277138963416770734827, 5.37915785401627898631541565252, 5.87528865673177748562799631708, 6.71689283504453592886645528269, 7.78459739864314372018344946247
