| 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
| \(L(1)\) |
\(\approx\) |
\(2.931265271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.931265271\) |
| \(L(\frac{3}{2})\) |
|
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
−12.63606651356499080629382469286, −11.49519150041699428074254799988, −10.44446913418299998873332249913, −9.489302751643217678452931395295, −7.69480844162426561675059666923, −6.69922177604601220464976850491, −5.76267162674386544228975097635, −4.86075960264171060465219091221, −3.47169965665101283778552642364, −2.57844828379127453678469692471,
2.57844828379127453678469692471, 3.47169965665101283778552642364, 4.86075960264171060465219091221, 5.76267162674386544228975097635, 6.69922177604601220464976850491, 7.69480844162426561675059666923, 9.489302751643217678452931395295, 10.44446913418299998873332249913, 11.49519150041699428074254799988, 12.63606651356499080629382469286
