| L(s) = 1 | − 2.55·2-s − 0.498·3-s + 4.54·4-s + 3.86·5-s + 1.27·6-s − 1.47·7-s − 6.50·8-s − 2.75·9-s − 9.88·10-s + 3.73·11-s − 2.26·12-s + 2.20·13-s + 3.77·14-s − 1.92·15-s + 7.55·16-s − 5.19·17-s + 7.03·18-s + 3.91·19-s + 17.5·20-s + 0.735·21-s − 9.55·22-s + 1.19·23-s + 3.24·24-s + 9.94·25-s − 5.63·26-s + 2.86·27-s − 6.70·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.287·3-s + 2.27·4-s + 1.72·5-s + 0.520·6-s − 0.558·7-s − 2.29·8-s − 0.917·9-s − 3.12·10-s + 1.12·11-s − 0.653·12-s + 0.610·13-s + 1.00·14-s − 0.497·15-s + 1.88·16-s − 1.25·17-s + 1.65·18-s + 0.897·19-s + 3.92·20-s + 0.160·21-s − 2.03·22-s + 0.248·23-s + 0.661·24-s + 1.98·25-s − 1.10·26-s + 0.551·27-s − 1.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 + 0.498T + 3T^{2} \) |
| 5 | \( 1 - 3.86T + 5T^{2} \) |
| 7 | \( 1 + 1.47T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 - 2.20T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 + 0.523T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 1.39T + 37T^{2} \) |
| 41 | \( 1 + 5.31T + 41T^{2} \) |
| 43 | \( 1 + 3.78T + 43T^{2} \) |
| 47 | \( 1 - 2.79T + 47T^{2} \) |
| 59 | \( 1 + 6.75T + 59T^{2} \) |
| 61 | \( 1 + 0.471T + 61T^{2} \) |
| 67 | \( 1 + 8.90T + 67T^{2} \) |
| 71 | \( 1 + 3.84T + 71T^{2} \) |
| 73 | \( 1 + 4.56T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 0.252T + 83T^{2} \) |
| 89 | \( 1 + 6.37T + 89T^{2} \) |
| 97 | \( 1 - 5.29T + 97T^{2} \) |
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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
−7.49363495430375866954685970140, −6.74248441417232187343369897324, −6.30506182542733638409434318349, −5.87931031695351143866590186007, −5.00731751445725848642239730793, −3.50019688466300104641221463013, −2.68320445142517102232508547520, −1.83763957869081904613517698819, −1.24886443509945034744890558224, 0,
1.24886443509945034744890558224, 1.83763957869081904613517698819, 2.68320445142517102232508547520, 3.50019688466300104641221463013, 5.00731751445725848642239730793, 5.87931031695351143866590186007, 6.30506182542733638409434318349, 6.74248441417232187343369897324, 7.49363495430375866954685970140
