L(s) = 1 | − 0.680·2-s − 2.33·3-s − 1.53·4-s + 3.24·5-s + 1.58·6-s + 2.40·8-s + 2.44·9-s − 2.20·10-s − 5.33·11-s + 3.58·12-s − 7.57·15-s + 1.43·16-s + 2.66·17-s − 1.66·18-s + 0.696·19-s − 4.99·20-s + 3.62·22-s + 8.96·23-s − 5.61·24-s + 5.53·25-s + 1.28·27-s − 5.28·29-s + 5.15·30-s + 5.45·31-s − 5.79·32-s + 12.4·33-s − 1.81·34-s + ⋯ |
L(s) = 1 | − 0.480·2-s − 1.34·3-s − 0.768·4-s + 1.45·5-s + 0.648·6-s + 0.850·8-s + 0.816·9-s − 0.698·10-s − 1.60·11-s + 1.03·12-s − 1.95·15-s + 0.359·16-s + 0.647·17-s − 0.392·18-s + 0.159·19-s − 1.11·20-s + 0.773·22-s + 1.86·23-s − 1.14·24-s + 1.10·25-s + 0.247·27-s − 0.981·29-s + 0.940·30-s + 0.980·31-s − 1.02·32-s + 2.16·33-s − 0.311·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8171991238\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8171991238\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.680T + 2T^{2} \) |
| 3 | \( 1 + 2.33T + 3T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 11 | \( 1 + 5.33T + 11T^{2} \) |
| 17 | \( 1 - 2.66T + 17T^{2} \) |
| 19 | \( 1 - 0.696T + 19T^{2} \) |
| 23 | \( 1 - 8.96T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 - 5.45T + 31T^{2} \) |
| 37 | \( 1 + 6.69T + 37T^{2} \) |
| 41 | \( 1 - 2.21T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 + 4.71T + 53T^{2} \) |
| 59 | \( 1 - 2.98T + 59T^{2} \) |
| 61 | \( 1 - 2.19T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 - 8.56T + 71T^{2} \) |
| 73 | \( 1 - 5.88T + 73T^{2} \) |
| 79 | \( 1 - 0.910T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 0.986T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.82249448328043399137461403539, −7.03201090950080935712417083307, −6.30191163683834353945289305097, −5.52499633722216320496708352657, −5.09038958527224858619547723294, −4.90948382526897738286084825534, −3.45950688085310795950050898193, −2.47475654179379871291795556303, −1.40795071410379295991725783962, −0.56000707070140132298633539503,
0.56000707070140132298633539503, 1.40795071410379295991725783962, 2.47475654179379871291795556303, 3.45950688085310795950050898193, 4.90948382526897738286084825534, 5.09038958527224858619547723294, 5.52499633722216320496708352657, 6.30191163683834353945289305097, 7.03201090950080935712417083307, 7.82249448328043399137461403539