| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 3·11-s + 4·13-s − 2·14-s + 16-s − 5·17-s + 6·19-s + 3·22-s − 3·23-s + 4·26-s − 2·28-s − 3·29-s + 2·31-s + 32-s − 5·34-s + 2·37-s + 6·38-s − 8·41-s + 2·43-s + 3·44-s − 3·46-s − 3·49-s + 4·52-s + 4·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.904·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.21·17-s + 1.37·19-s + 0.639·22-s − 0.625·23-s + 0.784·26-s − 0.377·28-s − 0.557·29-s + 0.359·31-s + 0.176·32-s − 0.857·34-s + 0.328·37-s + 0.973·38-s − 1.24·41-s + 0.304·43-s + 0.452·44-s − 0.442·46-s − 3/7·49-s + 0.554·52-s + 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.50130452436030, −13.99854603987110, −13.55187756704028, −13.25973812399030, −12.69307979639916, −12.01689130808258, −11.67899777846412, −11.20005019738209, −10.65802463670990, −9.994595390956753, −9.487704718214160, −8.980265039835895, −8.447915318333865, −7.753169344447913, −7.050359881468092, −6.684744486521527, −6.034346272219198, −5.800032824005737, −4.897015709320120, −4.332004468251720, −3.698929996788973, −3.321499437840272, −2.615143966715321, −1.729338183421133, −1.136566758595498, 0,
1.136566758595498, 1.729338183421133, 2.615143966715321, 3.321499437840272, 3.698929996788973, 4.332004468251720, 4.897015709320120, 5.800032824005737, 6.034346272219198, 6.684744486521527, 7.050359881468092, 7.753169344447913, 8.447915318333865, 8.980265039835895, 9.487704718214160, 9.994595390956753, 10.65802463670990, 11.20005019738209, 11.67899777846412, 12.01689130808258, 12.69307979639916, 13.25973812399030, 13.55187756704028, 13.99854603987110, 14.50130452436030
