L(s) = 1 | − 2.45·2-s + 4.04·4-s + 3.33·5-s + 3.69·7-s − 5.03·8-s − 8.20·10-s − 0.270·11-s − 9.08·14-s + 4.29·16-s − 4.04·17-s + 7.15·19-s + 13.5·20-s + 0.664·22-s − 2.79·23-s + 6.13·25-s + 14.9·28-s + 7.83·29-s + 2.14·31-s − 0.487·32-s + 9.93·34-s + 12.3·35-s + 4.37·37-s − 17.6·38-s − 16.8·40-s − 7.61·41-s − 3.89·43-s − 1.09·44-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 2.02·4-s + 1.49·5-s + 1.39·7-s − 1.78·8-s − 2.59·10-s − 0.0815·11-s − 2.42·14-s + 1.07·16-s − 0.980·17-s + 1.64·19-s + 3.02·20-s + 0.141·22-s − 0.583·23-s + 1.22·25-s + 2.82·28-s + 1.45·29-s + 0.385·31-s − 0.0861·32-s + 1.70·34-s + 2.08·35-s + 0.719·37-s − 2.85·38-s − 2.65·40-s − 1.18·41-s − 0.593·43-s − 0.165·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.224656723\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.224656723\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 5 | \( 1 - 3.33T + 5T^{2} \) |
| 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 + 0.270T + 11T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 19 | \( 1 - 7.15T + 19T^{2} \) |
| 23 | \( 1 + 2.79T + 23T^{2} \) |
| 29 | \( 1 - 7.83T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 + 7.61T + 41T^{2} \) |
| 43 | \( 1 + 3.89T + 43T^{2} \) |
| 47 | \( 1 - 0.877T + 47T^{2} \) |
| 53 | \( 1 - 6.83T + 53T^{2} \) |
| 59 | \( 1 - 3.40T + 59T^{2} \) |
| 61 | \( 1 - 3.06T + 61T^{2} \) |
| 67 | \( 1 + 5.00T + 67T^{2} \) |
| 71 | \( 1 + 7.59T + 71T^{2} \) |
| 73 | \( 1 + 0.405T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 6.98T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−9.490465272312245950841288219050, −8.682718701558320515758931592039, −8.168048449741246298959930604429, −7.26609683975038170934462683446, −6.48490756334629893357768950575, −5.55656650683225032278295317245, −4.67134955610527523759938122163, −2.72569184401813229672386857468, −1.88330121598362130862048006335, −1.11950046492394812314065934797,
1.11950046492394812314065934797, 1.88330121598362130862048006335, 2.72569184401813229672386857468, 4.67134955610527523759938122163, 5.55656650683225032278295317245, 6.48490756334629893357768950575, 7.26609683975038170934462683446, 8.168048449741246298959930604429, 8.682718701558320515758931592039, 9.490465272312245950841288219050