| L(s) = 1 | − 2.09·2-s + 2.40·4-s + 3.29·5-s + 7-s − 0.858·8-s − 6.90·10-s + 3.62·11-s − 2.16·13-s − 2.09·14-s − 3.01·16-s − 2.19·17-s − 6.66·19-s + 7.92·20-s − 7.61·22-s − 2.36·23-s + 5.82·25-s + 4.53·26-s + 2.40·28-s − 2.06·29-s − 2.19·31-s + 8.04·32-s + 4.60·34-s + 3.29·35-s − 10.1·37-s + 13.9·38-s − 2.82·40-s + 0.647·41-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 1.20·4-s + 1.47·5-s + 0.377·7-s − 0.303·8-s − 2.18·10-s + 1.09·11-s − 0.599·13-s − 0.561·14-s − 0.753·16-s − 0.531·17-s − 1.52·19-s + 1.77·20-s − 1.62·22-s − 0.493·23-s + 1.16·25-s + 0.889·26-s + 0.455·28-s − 0.382·29-s − 0.393·31-s + 1.42·32-s + 0.789·34-s + 0.556·35-s − 1.66·37-s + 2.27·38-s − 0.446·40-s + 0.101·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 5 | \( 1 - 3.29T + 5T^{2} \) |
| 11 | \( 1 - 3.62T + 11T^{2} \) |
| 13 | \( 1 + 2.16T + 13T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 + 6.66T + 19T^{2} \) |
| 23 | \( 1 + 2.36T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 0.647T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 - 8.59T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 - 0.592T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 0.664T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 3.24T + 73T^{2} \) |
| 79 | \( 1 - 4.01T + 79T^{2} \) |
| 83 | \( 1 - 3.40T + 83T^{2} \) |
| 89 | \( 1 + 6.31T + 89T^{2} \) |
| 97 | \( 1 + 0.0689T + 97T^{2} \) |
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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
−7.59023652776967271973834405214, −6.83176275880193191824582727212, −6.38760912046798310469491953870, −5.63854095132407408674556076604, −4.71658492410925193406748774582, −3.93995324189305407984113147747, −2.48843884456712230385783681781, −1.94105546729738287151163311347, −1.35544145584558757428642471806, 0,
1.35544145584558757428642471806, 1.94105546729738287151163311347, 2.48843884456712230385783681781, 3.93995324189305407984113147747, 4.71658492410925193406748774582, 5.63854095132407408674556076604, 6.38760912046798310469491953870, 6.83176275880193191824582727212, 7.59023652776967271973834405214
