| L(s) = 1 | − 2.60·2-s + 1.48·3-s + 4.77·4-s + 1.31·5-s − 3.85·6-s + 3.44·7-s − 7.20·8-s − 0.808·9-s − 3.41·10-s − 4.34·11-s + 7.06·12-s + 1.60·13-s − 8.96·14-s + 1.94·15-s + 9.21·16-s − 2.68·17-s + 2.10·18-s + 1.39·19-s + 6.25·20-s + 5.10·21-s + 11.2·22-s − 1.41·23-s − 10.6·24-s − 3.27·25-s − 4.17·26-s − 5.63·27-s + 16.4·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 0.854·3-s + 2.38·4-s + 0.586·5-s − 1.57·6-s + 1.30·7-s − 2.54·8-s − 0.269·9-s − 1.07·10-s − 1.30·11-s + 2.03·12-s + 0.444·13-s − 2.39·14-s + 0.501·15-s + 2.30·16-s − 0.650·17-s + 0.495·18-s + 0.319·19-s + 1.39·20-s + 1.11·21-s + 2.40·22-s − 0.295·23-s − 2.17·24-s − 0.655·25-s − 0.817·26-s − 1.08·27-s + 3.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 4003 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 - 1.48T + 3T^{2} \) |
| 5 | \( 1 - 1.31T + 5T^{2} \) |
| 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 1.60T + 13T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 19 | \( 1 - 1.39T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 + 8.88T + 31T^{2} \) |
| 37 | \( 1 - 6.69T + 37T^{2} \) |
| 41 | \( 1 + 3.03T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 1.90T + 47T^{2} \) |
| 53 | \( 1 - 9.71T + 53T^{2} \) |
| 59 | \( 1 - 1.71T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 0.441T + 67T^{2} \) |
| 71 | \( 1 + 7.97T + 71T^{2} \) |
| 73 | \( 1 + 5.45T + 73T^{2} \) |
| 79 | \( 1 - 2.73T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 0.127T + 97T^{2} \) |
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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
−8.111802111864049551805774218865, −7.68787903524340322341671238149, −7.24707949886030727942478871418, −5.85200848987354094430343916322, −5.52026855384474380740484229391, −4.05987210042307198591484786887, −2.77756980567878784299023734187, −2.15005571964981269129209904295, −1.55033423841109946845502159180, 0,
1.55033423841109946845502159180, 2.15005571964981269129209904295, 2.77756980567878784299023734187, 4.05987210042307198591484786887, 5.52026855384474380740484229391, 5.85200848987354094430343916322, 7.24707949886030727942478871418, 7.68787903524340322341671238149, 8.111802111864049551805774218865
