| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 2·13-s − 2·14-s + 16-s − 6·17-s + 6·23-s + 2·26-s − 2·28-s − 4·29-s + 32-s − 6·34-s + 10·37-s − 8·41-s − 2·43-s + 6·46-s − 2·47-s − 3·49-s + 2·52-s + 2·53-s − 2·56-s − 4·58-s + 14·61-s + 64-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 1.25·23-s + 0.392·26-s − 0.377·28-s − 0.742·29-s + 0.176·32-s − 1.02·34-s + 1.64·37-s − 1.24·41-s − 0.304·43-s + 0.884·46-s − 0.291·47-s − 3/7·49-s + 0.277·52-s + 0.274·53-s − 0.267·56-s − 0.525·58-s + 1.79·61-s + 1/8·64-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.30229196428434, −13.14767089399178, −12.76524027913840, −12.18158614063132, −11.50230240860387, −11.16824687630749, −10.92126121377260, −10.18277100919119, −9.667463077855282, −9.273905051977257, −8.650285159009959, −8.258987804729182, −7.573492821127069, −6.938925744933210, −6.575279646533036, −6.309693773645356, −5.513060475377837, −5.118642076519572, −4.524228379404657, −3.884291624514350, −3.553448533869142, −2.805800614920292, −2.401932368020356, −1.649556713716323, −0.8726571037675081, 0,
0.8726571037675081, 1.649556713716323, 2.401932368020356, 2.805800614920292, 3.553448533869142, 3.884291624514350, 4.524228379404657, 5.118642076519572, 5.513060475377837, 6.309693773645356, 6.575279646533036, 6.938925744933210, 7.573492821127069, 8.258987804729182, 8.650285159009959, 9.273905051977257, 9.667463077855282, 10.18277100919119, 10.92126121377260, 11.16824687630749, 11.50230240860387, 12.18158614063132, 12.76524027913840, 13.14767089399178, 13.30229196428434
