| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 3·11-s + 6·13-s + 2·14-s + 16-s − 3·17-s + 2·19-s − 3·22-s − 5·23-s + 6·26-s + 2·28-s − 29-s + 4·31-s + 32-s − 3·34-s − 12·37-s + 2·38-s + 12·43-s − 3·44-s − 5·46-s + 4·47-s − 3·49-s + 6·52-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 + 1.66·13-s + 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s − 0.639·22-s − 1.04·23-s + 1.17·26-s + 0.377·28-s − 0.185·29-s + 0.718·31-s + 0.176·32-s − 0.514·34-s − 1.97·37-s + 0.324·38-s + 1.82·43-s − 0.452·44-s − 0.737·46-s + 0.583·47-s − 3/7·49-s + 0.832·52-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 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−14.44887161076104, −14.01236007650751, −13.58271471105193, −13.35748930686845, −12.53644266861202, −12.18089222118738, −11.59570235343310, −11.00015721194399, −10.69600759322754, −10.32625392643723, −9.388237866203539, −8.888384118039742, −8.245433719429385, −7.880072466823746, −7.320453750214167, −6.570157975983381, −6.036313405370543, −5.607544151535303, −4.948041124349664, −4.406940283084075, −3.838440261582069, −3.204589316785210, −2.507901760520044, −1.767195099540999, −1.203155528577206, 0,
1.203155528577206, 1.767195099540999, 2.507901760520044, 3.204589316785210, 3.838440261582069, 4.406940283084075, 4.948041124349664, 5.607544151535303, 6.036313405370543, 6.570157975983381, 7.320453750214167, 7.880072466823746, 8.245433719429385, 8.888384118039742, 9.388237866203539, 10.32625392643723, 10.69600759322754, 11.00015721194399, 11.59570235343310, 12.18089222118738, 12.53644266861202, 13.35748930686845, 13.58271471105193, 14.01236007650751, 14.44887161076104
