| L(s) = 1 | + 2.23·2-s − 3.23·3-s + 3.00·4-s + 1.23·5-s − 7.23·6-s − 2.85·7-s + 2.23·8-s + 7.47·9-s + 2.76·10-s − 9.70·12-s − 13-s − 6.38·14-s − 4.00·15-s − 0.999·16-s + 2.38·17-s + 16.7·18-s + 0.618·19-s + 3.70·20-s + 9.23·21-s − 8·23-s − 7.23·24-s − 3.47·25-s − 2.23·26-s − 14.4·27-s − 8.56·28-s − 3.38·29-s − 8.94·30-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 1.86·3-s + 1.50·4-s + 0.552·5-s − 2.95·6-s − 1.07·7-s + 0.790·8-s + 2.49·9-s + 0.874·10-s − 2.80·12-s − 0.277·13-s − 1.70·14-s − 1.03·15-s − 0.249·16-s + 0.577·17-s + 3.93·18-s + 0.141·19-s + 0.829·20-s + 2.01·21-s − 1.66·23-s − 1.47·24-s − 0.694·25-s − 0.438·26-s − 2.78·27-s − 1.61·28-s − 0.628·29-s − 1.63·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 - 0.618T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 3.38T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 9.32T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 7.38T + 59T^{2} \) |
| 61 | \( 1 - 8.32T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 8.56T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 3.09T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 3.70T + 97T^{2} \) |
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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
−9.648598415644563765420543581111, −7.75086563255156057230837645454, −6.75155416947565726489220900064, −6.13479291426532709104498425031, −5.84910280677242657678599699767, −5.00677449675646004260050206939, −4.22661208996636518035825465443, −3.28197600963523948098296392950, −1.83752924113669288401899544466, 0,
1.83752924113669288401899544466, 3.28197600963523948098296392950, 4.22661208996636518035825465443, 5.00677449675646004260050206939, 5.84910280677242657678599699767, 6.13479291426532709104498425031, 6.75155416947565726489220900064, 7.75086563255156057230837645454, 9.648598415644563765420543581111
