L(s) = 1 | + 0.446·2-s − 3.01·3-s − 1.80·4-s − 2.85·5-s − 1.34·6-s + 4.83·7-s − 1.69·8-s + 6.11·9-s − 1.27·10-s − 1.74·11-s + 5.43·12-s + 2.02·13-s + 2.15·14-s + 8.63·15-s + 2.84·16-s + 17-s + 2.73·18-s + 2.33·19-s + 5.14·20-s − 14.6·21-s − 0.780·22-s − 4.72·23-s + 5.12·24-s + 3.16·25-s + 0.905·26-s − 9.41·27-s − 8.71·28-s + ⋯ |
L(s) = 1 | + 0.315·2-s − 1.74·3-s − 0.900·4-s − 1.27·5-s − 0.550·6-s + 1.82·7-s − 0.599·8-s + 2.03·9-s − 0.403·10-s − 0.527·11-s + 1.56·12-s + 0.562·13-s + 0.577·14-s + 2.22·15-s + 0.710·16-s + 0.242·17-s + 0.643·18-s + 0.534·19-s + 1.15·20-s − 3.18·21-s − 0.166·22-s − 0.985·23-s + 1.04·24-s + 0.633·25-s + 0.177·26-s − 1.81·27-s − 1.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 - 0.446T + 2T^{2} \) |
| 3 | \( 1 + 3.01T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 - 2.02T + 13T^{2} \) |
| 19 | \( 1 - 2.33T + 19T^{2} \) |
| 23 | \( 1 + 4.72T + 23T^{2} \) |
| 29 | \( 1 + 1.78T + 29T^{2} \) |
| 31 | \( 1 - 3.49T + 31T^{2} \) |
| 37 | \( 1 + 1.26T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 - 5.36T + 53T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 1.62T + 67T^{2} \) |
| 71 | \( 1 - 8.65T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 6.06T + 79T^{2} \) |
| 83 | \( 1 + 5.78T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−9.827223338517179728413019066851, −8.276842147641582999840512050378, −8.077772502219215317264325960098, −6.98039028847532409362292132213, −5.71690868829480211197800466727, −5.09987467672319916780987375513, −4.51678591039551637316219634453, −3.72520589161345596182241287308, −1.32080753287751811647151706440, 0,
1.32080753287751811647151706440, 3.72520589161345596182241287308, 4.51678591039551637316219634453, 5.09987467672319916780987375513, 5.71690868829480211197800466727, 6.98039028847532409362292132213, 8.077772502219215317264325960098, 8.276842147641582999840512050378, 9.827223338517179728413019066851