L(s) = 1 | + 1.33·2-s + 3-s − 0.229·4-s − 2.44·5-s + 1.33·6-s + 4.33·7-s − 2.96·8-s + 9-s − 3.24·10-s + 1.31·11-s − 0.229·12-s − 13-s + 5.76·14-s − 2.44·15-s − 3.48·16-s − 1.21·17-s + 1.33·18-s + 6.39·19-s + 0.559·20-s + 4.33·21-s + 1.75·22-s − 0.356·23-s − 2.96·24-s + 0.960·25-s − 1.33·26-s + 27-s − 0.993·28-s + ⋯ |
L(s) = 1 | + 0.940·2-s + 0.577·3-s − 0.114·4-s − 1.09·5-s + 0.543·6-s + 1.63·7-s − 1.04·8-s + 0.333·9-s − 1.02·10-s + 0.396·11-s − 0.0661·12-s − 0.277·13-s + 1.54·14-s − 0.630·15-s − 0.872·16-s − 0.294·17-s + 0.313·18-s + 1.46·19-s + 0.125·20-s + 0.945·21-s + 0.373·22-s − 0.0743·23-s − 0.605·24-s + 0.192·25-s − 0.260·26-s + 0.192·27-s − 0.187·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.321673517\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.321673517\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 - 1.33T + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 - 4.33T + 7T^{2} \) |
| 11 | \( 1 - 1.31T + 11T^{2} \) |
| 17 | \( 1 + 1.21T + 17T^{2} \) |
| 19 | \( 1 - 6.39T + 19T^{2} \) |
| 23 | \( 1 + 0.356T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 + 2.92T + 31T^{2} \) |
| 37 | \( 1 - 9.29T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 - 9.88T + 43T^{2} \) |
| 47 | \( 1 - 1.03T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 - 9.20T + 59T^{2} \) |
| 61 | \( 1 + 6.79T + 61T^{2} \) |
| 67 | \( 1 + 8.18T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 + 4.14T + 73T^{2} \) |
| 79 | \( 1 - 4.44T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.234153329453192462711803927935, −7.74279218193172023733212603462, −7.24133864361404194446251005533, −5.98155501794647828084285387520, −5.20710081731343737941525215407, −4.46256820235506159685596548293, −4.06220771089799373520446916754, −3.23326763990158068910833068741, −2.22594125352549712184241827006, −0.917025733197366551774691928985,
0.917025733197366551774691928985, 2.22594125352549712184241827006, 3.23326763990158068910833068741, 4.06220771089799373520446916754, 4.46256820235506159685596548293, 5.20710081731343737941525215407, 5.98155501794647828084285387520, 7.24133864361404194446251005533, 7.74279218193172023733212603462, 8.234153329453192462711803927935