| L(s) = 1 | + 1.37·2-s − 3-s − 0.105·4-s − 1.48·5-s − 1.37·6-s + 4.92·7-s − 2.89·8-s + 9-s − 2.04·10-s − 4.78·11-s + 0.105·12-s + 0.524·13-s + 6.78·14-s + 1.48·15-s − 3.77·16-s − 2.51·17-s + 1.37·18-s + 6.02·19-s + 0.156·20-s − 4.92·21-s − 6.58·22-s − 23-s + 2.89·24-s − 2.80·25-s + 0.722·26-s − 27-s − 0.519·28-s + ⋯ |
| L(s) = 1 | + 0.973·2-s − 0.577·3-s − 0.0527·4-s − 0.663·5-s − 0.561·6-s + 1.86·7-s − 1.02·8-s + 0.333·9-s − 0.645·10-s − 1.44·11-s + 0.0304·12-s + 0.145·13-s + 1.81·14-s + 0.382·15-s − 0.944·16-s − 0.610·17-s + 0.324·18-s + 1.38·19-s + 0.0349·20-s − 1.07·21-s − 1.40·22-s − 0.208·23-s + 0.591·24-s − 0.560·25-s + 0.141·26-s − 0.192·27-s − 0.0982·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.37T + 2T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 4.78T + 11T^{2} \) |
| 13 | \( 1 - 0.524T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 - 6.02T + 19T^{2} \) |
| 31 | \( 1 + 4.86T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 + 6.88T + 41T^{2} \) |
| 43 | \( 1 - 5.44T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 + 8.40T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 8.80T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 5.81T + 79T^{2} \) |
| 83 | \( 1 - 7.77T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 8.12T + 97T^{2} \) |
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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
−8.557067081653388444892753776295, −7.84220187960460105484590151876, −7.34904413044430295113897879977, −5.99566626711512335464381772501, −5.21688984158902518939262160145, −4.87113362194729630164746003281, −4.09505285939631164223516319076, −3.03024647219341291817864318798, −1.70173509313427322505706972736, 0,
1.70173509313427322505706972736, 3.03024647219341291817864318798, 4.09505285939631164223516319076, 4.87113362194729630164746003281, 5.21688984158902518939262160145, 5.99566626711512335464381772501, 7.34904413044430295113897879977, 7.84220187960460105484590151876, 8.557067081653388444892753776295
