| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 3·11-s + 4·13-s − 4·14-s + 16-s − 4·19-s + 3·22-s − 4·26-s + 4·28-s − 6·29-s − 31-s − 32-s + 4·37-s + 4·38-s − 6·41-s − 5·43-s − 3·44-s + 12·47-s + 9·49-s + 4·52-s − 6·53-s − 4·56-s + 6·58-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 0.904·11-s + 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.917·19-s + 0.639·22-s − 0.784·26-s + 0.755·28-s − 1.11·29-s − 0.179·31-s − 0.176·32-s + 0.657·37-s + 0.648·38-s − 0.937·41-s − 0.762·43-s − 0.452·44-s + 1.75·47-s + 9/7·49-s + 0.554·52-s − 0.824·53-s − 0.534·56-s + 0.787·58-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.79958049511228, −14.03522188549511, −13.74961213366744, −13.05217016538655, −12.57426267000720, −11.96979419884494, −11.27370456065962, −10.96785192509307, −10.76024586234364, −10.04818625980945, −9.412652630406371, −8.813289531333867, −8.226965137139946, −8.120253135112467, −7.465292526409908, −6.870086259591295, −6.180939498693304, −5.557070126847938, −5.125641145375468, −4.359083622778933, −3.818476725303965, −2.974229747457339, −2.164789216984405, −1.726701389629759, −0.9968285847710098, 0,
0.9968285847710098, 1.726701389629759, 2.164789216984405, 2.974229747457339, 3.818476725303965, 4.359083622778933, 5.125641145375468, 5.557070126847938, 6.180939498693304, 6.870086259591295, 7.465292526409908, 8.120253135112467, 8.226965137139946, 8.813289531333867, 9.412652630406371, 10.04818625980945, 10.76024586234364, 10.96785192509307, 11.27370456065962, 11.96979419884494, 12.57426267000720, 13.05217016538655, 13.74961213366744, 14.03522188549511, 14.79958049511228
