| L(s) = 1 | + 2.02·2-s + 2.09·4-s − 1.44·5-s + 0.381·7-s + 0.196·8-s − 2.91·10-s − 0.280·11-s − 4.00·13-s + 0.771·14-s − 3.79·16-s − 2.60·17-s − 3.86·19-s − 3.02·20-s − 0.568·22-s − 0.698·23-s − 2.92·25-s − 8.10·26-s + 0.799·28-s − 1.62·29-s + 9.73·31-s − 8.07·32-s − 5.26·34-s − 0.549·35-s + 4.41·37-s − 7.83·38-s − 0.283·40-s + 0.0157·41-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.04·4-s − 0.644·5-s + 0.144·7-s + 0.0696·8-s − 0.922·10-s − 0.0846·11-s − 1.11·13-s + 0.206·14-s − 0.948·16-s − 0.631·17-s − 0.887·19-s − 0.675·20-s − 0.121·22-s − 0.145·23-s − 0.584·25-s − 1.58·26-s + 0.151·28-s − 0.302·29-s + 1.74·31-s − 1.42·32-s − 0.903·34-s − 0.0928·35-s + 0.726·37-s − 1.27·38-s − 0.0448·40-s + 0.00246·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 - 0.381T + 7T^{2} \) |
| 11 | \( 1 + 0.280T + 11T^{2} \) |
| 13 | \( 1 + 4.00T + 13T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 + 3.86T + 19T^{2} \) |
| 23 | \( 1 + 0.698T + 23T^{2} \) |
| 29 | \( 1 + 1.62T + 29T^{2} \) |
| 31 | \( 1 - 9.73T + 31T^{2} \) |
| 37 | \( 1 - 4.41T + 37T^{2} \) |
| 41 | \( 1 - 0.0157T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 7.71T + 47T^{2} \) |
| 53 | \( 1 - 4.91T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 6.97T + 61T^{2} \) |
| 67 | \( 1 + 2.97T + 67T^{2} \) |
| 71 | \( 1 + 7.28T + 71T^{2} \) |
| 73 | \( 1 + 0.165T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 8.75T + 97T^{2} \) |
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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
−8.479878981655210612916894420848, −7.83244423913283923846952607203, −6.80505549665223109564966945004, −6.29724148586050284540542212332, −5.19772799718701366824763029485, −4.60324508475535505095904208711, −3.96201630126956934768855626961, −2.96946367664736751302798111679, −2.09603170766045734500616107289, 0,
2.09603170766045734500616107289, 2.96946367664736751302798111679, 3.96201630126956934768855626961, 4.60324508475535505095904208711, 5.19772799718701366824763029485, 6.29724148586050284540542212332, 6.80505549665223109564966945004, 7.83244423913283923846952607203, 8.479878981655210612916894420848
