L(s) = 1 | − 2.53·2-s + 4.41·4-s − 0.467·5-s − 3.22·7-s − 6.10·8-s + 1.18·10-s + 3.10·11-s − 2.18·13-s + 8.17·14-s + 6.63·16-s − 3·17-s + 0.0418·19-s − 2.06·20-s − 7.86·22-s − 6.10·23-s − 4.78·25-s + 5.53·26-s − 14.2·28-s − 6.57·29-s − 6.22·31-s − 4.59·32-s + 7.59·34-s + 1.50·35-s + 3.59·37-s − 0.106·38-s + 2.85·40-s − 7.70·41-s + ⋯ |
L(s) = 1 | − 1.79·2-s + 2.20·4-s − 0.209·5-s − 1.21·7-s − 2.15·8-s + 0.374·10-s + 0.936·11-s − 0.605·13-s + 2.18·14-s + 1.65·16-s − 0.727·17-s + 0.00961·19-s − 0.461·20-s − 1.67·22-s − 1.27·23-s − 0.956·25-s + 1.08·26-s − 2.69·28-s − 1.22·29-s − 1.11·31-s − 0.812·32-s + 1.30·34-s + 0.255·35-s + 0.591·37-s − 0.0172·38-s + 0.451·40-s − 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 5 | \( 1 + 0.467T + 5T^{2} \) |
| 7 | \( 1 + 3.22T + 7T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 0.0418T + 19T^{2} \) |
| 23 | \( 1 + 6.10T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 + 6.22T + 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 + 0.588T + 43T^{2} \) |
| 47 | \( 1 - 9.66T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 - 8.53T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 8.23T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 1.50T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−11.42460642410637826890522005543, −10.31564173334467064296540622658, −9.541383471047957055748250942967, −8.992831124463235558092316781552, −7.77789329088139499425888908988, −6.91356097673695218347818511037, −6.03587855749369941395695055753, −3.74567657895574150041096517072, −2.08831159546667970910620055785, 0,
2.08831159546667970910620055785, 3.74567657895574150041096517072, 6.03587855749369941395695055753, 6.91356097673695218347818511037, 7.77789329088139499425888908988, 8.992831124463235558092316781552, 9.541383471047957055748250942967, 10.31564173334467064296540622658, 11.42460642410637826890522005543