L(s) = 1 | − 2.70·2-s − 1.17·3-s + 5.34·4-s − 1.17·5-s + 3.17·6-s − 7-s − 9.04·8-s − 1.63·9-s + 3.17·10-s + 3.70·11-s − 6.24·12-s + 4.34·13-s + 2.70·14-s + 1.36·15-s + 13.8·16-s + 3.17·17-s + 4.41·18-s + 5.26·19-s − 6.24·20-s + 1.17·21-s − 10.0·22-s + 23-s + 10.5·24-s − 3.63·25-s − 11.7·26-s + 5.41·27-s − 5.34·28-s + ⋯ |
L(s) = 1 | − 1.91·2-s − 0.675·3-s + 2.67·4-s − 0.523·5-s + 1.29·6-s − 0.377·7-s − 3.19·8-s − 0.543·9-s + 1.00·10-s + 1.11·11-s − 1.80·12-s + 1.20·13-s + 0.724·14-s + 0.353·15-s + 3.45·16-s + 0.768·17-s + 1.04·18-s + 1.20·19-s − 1.39·20-s + 0.255·21-s − 2.14·22-s + 0.208·23-s + 2.16·24-s − 0.726·25-s − 2.30·26-s + 1.04·27-s − 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3699755096\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3699755096\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 + 1.17T + 3T^{2} \) |
| 5 | \( 1 + 1.17T + 5T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 3.17T + 17T^{2} \) |
| 19 | \( 1 - 5.26T + 19T^{2} \) |
| 29 | \( 1 - 0.630T + 29T^{2} \) |
| 31 | \( 1 - 9.32T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 + 8.49T + 43T^{2} \) |
| 47 | \( 1 - 6.09T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 + 8.40T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 - 9.86T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 4.38T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 - 8.58T + 97T^{2} \) |
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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
−11.97052814488866233963419296681, −11.74293339943825567946657654225, −10.77269785100173446229521182710, −9.763958950499650302845123471036, −8.814657773032786599879890482509, −7.927265886305811418263577531455, −6.72611285528751165576175046847, −5.89368929581306666774079237618, −3.30320249699412220051216951037, −1.02437721082044214907661604132,
1.02437721082044214907661604132, 3.30320249699412220051216951037, 5.89368929581306666774079237618, 6.72611285528751165576175046847, 7.927265886305811418263577531455, 8.814657773032786599879890482509, 9.763958950499650302845123471036, 10.77269785100173446229521182710, 11.74293339943825567946657654225, 11.97052814488866233963419296681