| L(s) = 1 | + 2-s + 2.77·3-s + 4-s − 0.696·5-s + 2.77·6-s − 4.01·7-s + 8-s + 4.67·9-s − 0.696·10-s − 4.86·11-s + 2.77·12-s − 3.45·13-s − 4.01·14-s − 1.92·15-s + 16-s + 3.58·17-s + 4.67·18-s − 5.73·19-s − 0.696·20-s − 11.1·21-s − 4.86·22-s − 3.06·23-s + 2.77·24-s − 4.51·25-s − 3.45·26-s + 4.63·27-s − 4.01·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.59·3-s + 0.5·4-s − 0.311·5-s + 1.13·6-s − 1.51·7-s + 0.353·8-s + 1.55·9-s − 0.220·10-s − 1.46·11-s + 0.799·12-s − 0.957·13-s − 1.07·14-s − 0.497·15-s + 0.250·16-s + 0.869·17-s + 1.10·18-s − 1.31·19-s − 0.155·20-s − 2.42·21-s − 1.03·22-s − 0.639·23-s + 0.565·24-s − 0.903·25-s − 0.677·26-s + 0.891·27-s − 0.758·28-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 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 + 0.696T + 5T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 + 4.86T + 11T^{2} \) |
| 13 | \( 1 + 3.45T + 13T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 - 6.90T + 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 + 8.33T + 47T^{2} \) |
| 53 | \( 1 - 6.20T + 53T^{2} \) |
| 59 | \( 1 + 6.49T + 59T^{2} \) |
| 61 | \( 1 + 1.03T + 61T^{2} \) |
| 67 | \( 1 + 0.657T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + 9.67T + 83T^{2} \) |
| 89 | \( 1 + 6.22T + 89T^{2} \) |
| 97 | \( 1 + 4.99T + 97T^{2} \) |
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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
−8.041979733203597128438990452778, −7.46531696258273836609496314799, −6.70268227639196073936914528239, −5.89240876667625046952863073363, −4.87620400805602718371752297205, −4.00920894125741431585727789061, −3.26443115038822969651023213413, −2.74162715507112329233694333920, −2.07090219666600191623936126444, 0,
2.07090219666600191623936126444, 2.74162715507112329233694333920, 3.26443115038822969651023213413, 4.00920894125741431585727789061, 4.87620400805602718371752297205, 5.89240876667625046952863073363, 6.70268227639196073936914528239, 7.46531696258273836609496314799, 8.041979733203597128438990452778
