| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 12-s + 13-s − 14-s + 16-s − 5·17-s − 18-s + 5·19-s − 21-s + 23-s + 24-s − 5·25-s − 26-s − 27-s + 28-s − 9·29-s − 5·31-s − 32-s + 5·34-s + 36-s − 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 1.14·19-s − 0.218·21-s + 0.208·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s − 1.67·29-s − 0.898·31-s − 0.176·32-s + 0.857·34-s + 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.02042931506352, −13.38965505142272, −13.24619761218577, −12.54502432374289, −11.84479456496137, −11.59644608561732, −11.18525722515065, −10.76735174158531, −10.12867892377833, −9.721974822089536, −9.218620425662227, −8.672836993648774, −8.242878449201115, −7.592114771639656, −7.062539607567995, −6.841577846629140, −5.994668097688107, −5.603183947656009, −5.068660082574814, −4.483413763506892, −3.656131825636400, −3.329133787534726, −2.337453073922629, −1.662251217178524, −1.381088906164290, 0, 0,
1.381088906164290, 1.662251217178524, 2.337453073922629, 3.329133787534726, 3.656131825636400, 4.483413763506892, 5.068660082574814, 5.603183947656009, 5.994668097688107, 6.841577846629140, 7.062539607567995, 7.592114771639656, 8.242878449201115, 8.672836993648774, 9.218620425662227, 9.721974822089536, 10.12867892377833, 10.76735174158531, 11.18525722515065, 11.59644608561732, 11.84479456496137, 12.54502432374289, 13.24619761218577, 13.38965505142272, 14.02042931506352
