| L(s) = 1 | + 2-s + 4-s + 8-s − 2·13-s + 16-s − 6·17-s + 4·19-s + 8·23-s − 2·26-s − 2·29-s − 31-s + 32-s − 6·34-s − 10·37-s + 4·38-s + 6·41-s − 8·43-s + 8·46-s − 8·47-s − 7·49-s − 2·52-s − 6·53-s − 2·58-s + 12·59-s − 6·61-s − 62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 1.66·23-s − 0.392·26-s − 0.371·29-s − 0.179·31-s + 0.176·32-s − 1.02·34-s − 1.64·37-s + 0.648·38-s + 0.937·41-s − 1.21·43-s + 1.17·46-s − 1.16·47-s − 49-s − 0.277·52-s − 0.824·53-s − 0.262·58-s + 1.56·59-s − 0.768·61-s − 0.127·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.11822337290834, −15.98235074334685, −15.13618073881321, −14.82246363324919, −14.22792148516203, −13.52486873113799, −13.11871820750723, −12.65660003537967, −11.92823935733665, −11.30544553009214, −11.03173421817012, −10.20788017079349, −9.578844591943865, −8.947552296394550, −8.340165931939346, −7.473631943984117, −6.938197031630959, −6.525081261495663, −5.577954032023890, −4.973714760288934, −4.577442464572103, −3.567847491935261, −3.034755553522961, −2.189796099505077, −1.352817878928555, 0,
1.352817878928555, 2.189796099505077, 3.034755553522961, 3.567847491935261, 4.577442464572103, 4.973714760288934, 5.577954032023890, 6.525081261495663, 6.938197031630959, 7.473631943984117, 8.340165931939346, 8.947552296394550, 9.578844591943865, 10.20788017079349, 11.03173421817012, 11.30544553009214, 11.92823935733665, 12.65660003537967, 13.11871820750723, 13.52486873113799, 14.22792148516203, 14.82246363324919, 15.13618073881321, 15.98235074334685, 16.11822337290834
