L(s) = 1 | − 2-s − 4-s − 3·5-s − 4·7-s + 3·8-s − 3·9-s + 3·10-s − 5·11-s − 5·13-s + 4·14-s − 16-s − 6·17-s + 3·18-s − 2·19-s + 3·20-s + 5·22-s − 5·23-s + 4·25-s + 5·26-s + 4·28-s − 6·29-s − 31-s − 5·32-s + 6·34-s + 12·35-s + 3·36-s − 10·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.34·5-s − 1.51·7-s + 1.06·8-s − 9-s + 0.948·10-s − 1.50·11-s − 1.38·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.458·19-s + 0.670·20-s + 1.06·22-s − 1.04·23-s + 4/5·25-s + 0.980·26-s + 0.755·28-s − 1.11·29-s − 0.179·31-s − 0.883·32-s + 1.02·34-s + 2.02·35-s + 1/2·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22481 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 22481 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.31412744680317, −15.84506486430773, −15.40089152022480, −14.88564954708737, −14.13729513875034, −13.46695896171871, −13.04992604817202, −12.55223384962710, −12.00169070414373, −11.31201966340766, −10.75600056824182, −10.16598770995452, −9.734495038688310, −9.088355321211834, −8.431163087103359, −8.139967942913953, −7.456894954199839, −6.956473617631122, −6.289762735567401, −5.142862181184030, −5.052613258470793, −3.939008856989872, −3.565908922486521, −2.684976593361542, −2.061822175796283, 0, 0, 0,
2.061822175796283, 2.684976593361542, 3.565908922486521, 3.939008856989872, 5.052613258470793, 5.142862181184030, 6.289762735567401, 6.956473617631122, 7.456894954199839, 8.139967942913953, 8.431163087103359, 9.088355321211834, 9.734495038688310, 10.16598770995452, 10.75600056824182, 11.31201966340766, 12.00169070414373, 12.55223384962710, 13.04992604817202, 13.46695896171871, 14.13729513875034, 14.88564954708737, 15.40089152022480, 15.84506486430773, 16.31412744680317