| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s − 2·7-s − 8-s + 9-s + 10-s − 2·11-s + 2·12-s − 2·13-s + 2·14-s − 2·15-s + 16-s − 6·17-s − 18-s − 4·19-s − 20-s − 4·21-s + 2·22-s − 6·23-s − 2·24-s + 25-s + 2·26-s − 4·27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.577·12-s − 0.554·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.872·21-s + 0.426·22-s − 1.25·23-s − 0.408·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.93178954082821, −13.53826391557285, −13.01867385776230, −12.69783746860416, −11.92343091784212, −11.59285192069050, −10.94864476316459, −10.33678396187354, −10.04300365818705, −9.406350210780615, −9.037431968781588, −8.437694470471743, −8.189662685452511, −7.667171950022797, −7.075777597680356, −6.576560727142735, −6.095743290200689, −5.332951875596424, −4.564805261680953, −3.923837660575088, −3.546606345692867, −2.657532633930067, −2.361687168327502, −1.943227818490167, −0.6393430937637406, 0,
0.6393430937637406, 1.943227818490167, 2.361687168327502, 2.657532633930067, 3.546606345692867, 3.923837660575088, 4.564805261680953, 5.332951875596424, 6.095743290200689, 6.576560727142735, 7.075777597680356, 7.667171950022797, 8.189662685452511, 8.437694470471743, 9.037431968781588, 9.406350210780615, 10.04300365818705, 10.33678396187354, 10.94864476316459, 11.59285192069050, 11.92343091784212, 12.69783746860416, 13.01867385776230, 13.53826391557285, 13.93178954082821
