| L(s) = 1 | − 2-s − 2.83·3-s + 4-s − 3.27·5-s + 2.83·6-s − 1.58·7-s − 8-s + 5.02·9-s + 3.27·10-s + 2.07·11-s − 2.83·12-s − 0.763·13-s + 1.58·14-s + 9.26·15-s + 16-s + 4.68·17-s − 5.02·18-s + 6.23·19-s − 3.27·20-s + 4.49·21-s − 2.07·22-s − 5.02·23-s + 2.83·24-s + 5.69·25-s + 0.763·26-s − 5.73·27-s − 1.58·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.63·3-s + 0.5·4-s − 1.46·5-s + 1.15·6-s − 0.599·7-s − 0.353·8-s + 1.67·9-s + 1.03·10-s + 0.624·11-s − 0.817·12-s − 0.211·13-s + 0.423·14-s + 2.39·15-s + 0.250·16-s + 1.13·17-s − 1.18·18-s + 1.43·19-s − 0.731·20-s + 0.980·21-s − 0.441·22-s − 1.04·23-s + 0.578·24-s + 1.13·25-s + 0.149·26-s − 1.10·27-s − 0.299·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3519886799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3519886799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 2.83T + 3T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 - 6.23T + 19T^{2} \) |
| 23 | \( 1 + 5.02T + 23T^{2} \) |
| 29 | \( 1 - 7.85T + 29T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 5.08T + 41T^{2} \) |
| 43 | \( 1 + 6.46T + 43T^{2} \) |
| 47 | \( 1 - 5.88T + 47T^{2} \) |
| 53 | \( 1 + 1.72T + 53T^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 - 8.63T + 61T^{2} \) |
| 67 | \( 1 + 7.02T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 4.45T + 73T^{2} \) |
| 79 | \( 1 + 0.488T + 79T^{2} \) |
| 83 | \( 1 + 1.42T + 83T^{2} \) |
| 89 | \( 1 - 3.62T + 89T^{2} \) |
| 97 | \( 1 - 1.21T + 97T^{2} \) |
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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
−8.251729915182543650705292193417, −7.64839341652363668111510525083, −6.96013083110834305166223253654, −6.41873536993263956135016528625, −5.55796481569814727199636466112, −4.82602968957421142411532177959, −3.82984560796445802415273391979, −3.15525198390936650208306751607, −1.34356499269233130971687567543, −0.45797368893918625354413881510,
0.45797368893918625354413881510, 1.34356499269233130971687567543, 3.15525198390936650208306751607, 3.82984560796445802415273391979, 4.82602968957421142411532177959, 5.55796481569814727199636466112, 6.41873536993263956135016528625, 6.96013083110834305166223253654, 7.64839341652363668111510525083, 8.251729915182543650705292193417
