| L(s) = 1 | − 2.09·2-s − 3-s + 2.37·4-s + 4.26·5-s + 2.09·6-s + 3.25·7-s − 0.779·8-s + 9-s − 8.92·10-s − 3.52·11-s − 2.37·12-s − 5.02·13-s − 6.79·14-s − 4.26·15-s − 3.11·16-s + 0.0201·17-s − 2.09·18-s − 0.844·19-s + 10.1·20-s − 3.25·21-s + 7.37·22-s − 23-s + 0.779·24-s + 13.2·25-s + 10.4·26-s − 27-s + 7.71·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s − 0.577·3-s + 1.18·4-s + 1.90·5-s + 0.853·6-s + 1.22·7-s − 0.275·8-s + 0.333·9-s − 2.82·10-s − 1.06·11-s − 0.684·12-s − 1.39·13-s − 1.81·14-s − 1.10·15-s − 0.778·16-s + 0.00487·17-s − 0.492·18-s − 0.193·19-s + 2.26·20-s − 0.709·21-s + 1.57·22-s − 0.208·23-s + 0.159·24-s + 2.64·25-s + 2.05·26-s − 0.192·27-s + 1.45·28-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
| \(L(1)\) |
\(\approx\) |
\(1.013736450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013736450\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.09T + 2T^{2} \) |
| 5 | \( 1 - 4.26T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 + 5.02T + 13T^{2} \) |
| 17 | \( 1 - 0.0201T + 17T^{2} \) |
| 19 | \( 1 + 0.844T + 19T^{2} \) |
| 31 | \( 1 - 9.32T + 31T^{2} \) |
| 37 | \( 1 + 8.07T + 37T^{2} \) |
| 41 | \( 1 - 2.76T + 41T^{2} \) |
| 43 | \( 1 + 0.721T + 43T^{2} \) |
| 47 | \( 1 - 5.90T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 2.90T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 1.28T + 73T^{2} \) |
| 79 | \( 1 - 7.24T + 79T^{2} \) |
| 83 | \( 1 - 6.40T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 5.01T + 97T^{2} \) |
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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
−9.270866997318350187003135622758, −8.494278362537091597434758244800, −7.73459558072590633207621875015, −6.98007081554250220148313513336, −6.11175263987073395093799973156, −5.08911136658317318763603094622, −4.84022244040968579768411788143, −2.43069816472575590400181684674, −2.03795691814397297252520179955, −0.880718704890643490180811499365,
0.880718704890643490180811499365, 2.03795691814397297252520179955, 2.43069816472575590400181684674, 4.84022244040968579768411788143, 5.08911136658317318763603094622, 6.11175263987073395093799973156, 6.98007081554250220148313513336, 7.73459558072590633207621875015, 8.494278362537091597434758244800, 9.270866997318350187003135622758
