| L(s) = 1 | + 2-s − 4-s − 3·7-s − 3·8-s + 2·13-s − 3·14-s − 16-s + 3·17-s + 19-s − 23-s + 2·26-s + 3·28-s + 6·29-s + 4·31-s + 5·32-s + 3·34-s − 37-s + 38-s − 5·41-s + 4·43-s − 46-s − 3·47-s + 2·49-s − 2·52-s − 10·53-s + 9·56-s + 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.13·7-s − 1.06·8-s + 0.554·13-s − 0.801·14-s − 1/4·16-s + 0.727·17-s + 0.229·19-s − 0.208·23-s + 0.392·26-s + 0.566·28-s + 1.11·29-s + 0.718·31-s + 0.883·32-s + 0.514·34-s − 0.164·37-s + 0.162·38-s − 0.780·41-s + 0.609·43-s − 0.147·46-s − 0.437·47-s + 2/7·49-s − 0.277·52-s − 1.37·53-s + 1.20·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680293447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680293447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−15.26179246690525, −14.59874204990671, −14.01588870171116, −13.71589087380696, −13.17836755754987, −12.64330639449460, −12.16013714538357, −11.83477630611479, −10.92139093034775, −10.32841455926079, −9.765951889166425, −9.336866656648135, −8.757437139453650, −8.111067151987856, −7.537951408988087, −6.542663594254789, −6.265093577279235, −5.756661651481857, −4.869097654201256, −4.535842265997672, −3.506338790886375, −3.331933702594385, −2.639488147504717, −1.418302580495840, −0.4716653862781743,
0.4716653862781743, 1.418302580495840, 2.639488147504717, 3.331933702594385, 3.506338790886375, 4.535842265997672, 4.869097654201256, 5.756661651481857, 6.265093577279235, 6.542663594254789, 7.537951408988087, 8.111067151987856, 8.757437139453650, 9.336866656648135, 9.765951889166425, 10.32841455926079, 10.92139093034775, 11.83477630611479, 12.16013714538357, 12.64330639449460, 13.17836755754987, 13.71589087380696, 14.01588870171116, 14.59874204990671, 15.26179246690525
