| L(s) = 1 | + 2-s − 1.74·3-s + 4-s − 3.10·5-s − 1.74·6-s + 3.74·7-s + 8-s + 0.0511·9-s − 3.10·10-s − 1.53·11-s − 1.74·12-s − 1.75·13-s + 3.74·14-s + 5.42·15-s + 16-s + 0.774·17-s + 0.0511·18-s + 6.27·19-s − 3.10·20-s − 6.53·21-s − 1.53·22-s − 7.26·23-s − 1.74·24-s + 4.65·25-s − 1.75·26-s + 5.15·27-s + 3.74·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.00·3-s + 0.5·4-s − 1.38·5-s − 0.713·6-s + 1.41·7-s + 0.353·8-s + 0.0170·9-s − 0.982·10-s − 0.462·11-s − 0.504·12-s − 0.487·13-s + 1.00·14-s + 1.40·15-s + 0.250·16-s + 0.187·17-s + 0.0120·18-s + 1.44·19-s − 0.694·20-s − 1.42·21-s − 0.326·22-s − 1.51·23-s − 0.356·24-s + 0.930·25-s − 0.344·26-s + 0.991·27-s + 0.707·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 + 3.10T + 5T^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 + 1.75T + 13T^{2} \) |
| 17 | \( 1 - 0.774T + 17T^{2} \) |
| 19 | \( 1 - 6.27T + 19T^{2} \) |
| 23 | \( 1 + 7.26T + 23T^{2} \) |
| 29 | \( 1 - 2.50T + 29T^{2} \) |
| 31 | \( 1 + 0.103T + 31T^{2} \) |
| 37 | \( 1 + 4.16T + 37T^{2} \) |
| 41 | \( 1 + 8.62T + 41T^{2} \) |
| 43 | \( 1 - 6.29T + 43T^{2} \) |
| 47 | \( 1 + 0.664T + 47T^{2} \) |
| 53 | \( 1 + 2.26T + 53T^{2} \) |
| 59 | \( 1 - 0.525T + 59T^{2} \) |
| 61 | \( 1 - 0.0671T + 61T^{2} \) |
| 67 | \( 1 - 7.07T + 67T^{2} \) |
| 71 | \( 1 - 2.88T + 71T^{2} \) |
| 73 | \( 1 + 4.41T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 8.63T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 4.71T + 97T^{2} \) |
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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
−7.998752744204035865126343745629, −7.36818850215212941363886815473, −6.57996863918565607956045610676, −5.42446118020238201471042600953, −5.21177026040335506369105308929, −4.41464946469097121667786000318, −3.68621031287499806070370218116, −2.62434819483754436627306323677, −1.31488602830377365938674703721, 0,
1.31488602830377365938674703721, 2.62434819483754436627306323677, 3.68621031287499806070370218116, 4.41464946469097121667786000318, 5.21177026040335506369105308929, 5.42446118020238201471042600953, 6.57996863918565607956045610676, 7.36818850215212941363886815473, 7.998752744204035865126343745629
