L(s) = 1 | − 0.131·2-s − 1.12·3-s − 1.98·4-s + 1.29·5-s + 0.146·6-s + 2.28·7-s + 0.521·8-s − 1.74·9-s − 0.170·10-s + 5.69·11-s + 2.22·12-s − 0.913·13-s − 0.299·14-s − 1.45·15-s + 3.89·16-s − 8.00·17-s + 0.228·18-s − 5.65·19-s − 2.57·20-s − 2.56·21-s − 0.745·22-s − 4.08·23-s − 0.584·24-s − 3.31·25-s + 0.119·26-s + 5.31·27-s − 4.53·28-s + ⋯ |
L(s) = 1 | − 0.0926·2-s − 0.646·3-s − 0.991·4-s + 0.580·5-s + 0.0599·6-s + 0.864·7-s + 0.184·8-s − 0.581·9-s − 0.0538·10-s + 1.71·11-s + 0.641·12-s − 0.253·13-s − 0.0801·14-s − 0.375·15-s + 0.974·16-s − 1.94·17-s + 0.0538·18-s − 1.29·19-s − 0.575·20-s − 0.559·21-s − 0.159·22-s − 0.852·23-s − 0.119·24-s − 0.662·25-s + 0.0234·26-s + 1.02·27-s − 0.857·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2671 | \( 1 + T \) |
good | 2 | \( 1 + 0.131T + 2T^{2} \) |
| 3 | \( 1 + 1.12T + 3T^{2} \) |
| 5 | \( 1 - 1.29T + 5T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 - 5.69T + 11T^{2} \) |
| 13 | \( 1 + 0.913T + 13T^{2} \) |
| 17 | \( 1 + 8.00T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 4.08T + 23T^{2} \) |
| 29 | \( 1 - 7.63T + 29T^{2} \) |
| 31 | \( 1 + 1.92T + 31T^{2} \) |
| 37 | \( 1 - 7.31T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 8.04T + 43T^{2} \) |
| 47 | \( 1 - 8.52T + 47T^{2} \) |
| 53 | \( 1 + 0.0500T + 53T^{2} \) |
| 59 | \( 1 + 8.80T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 0.262T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 3.23T + 79T^{2} \) |
| 83 | \( 1 - 4.16T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 0.149T + 97T^{2} \) |
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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
−8.650539732099069400496099878191, −7.958800280979869521331708519961, −6.55268088261731280933583398482, −6.27873206522566843967534847837, −5.31852500245738342279429831804, −4.37948318492014906624649714771, −4.13828640435147936920758523742, −2.43214837531386334032289042870, −1.38629446771461180875041054322, 0,
1.38629446771461180875041054322, 2.43214837531386334032289042870, 4.13828640435147936920758523742, 4.37948318492014906624649714771, 5.31852500245738342279429831804, 6.27873206522566843967534847837, 6.55268088261731280933583398482, 7.958800280979869521331708519961, 8.650539732099069400496099878191