| L(s) = 1 | − 2.72·2-s − 1.76·3-s + 5.40·4-s + 0.469·5-s + 4.81·6-s + 1.03·7-s − 9.28·8-s + 0.123·9-s − 1.27·10-s − 1.84·11-s − 9.56·12-s − 0.700·13-s − 2.82·14-s − 0.830·15-s + 14.4·16-s + 0.793·17-s − 0.336·18-s + 3.65·19-s + 2.54·20-s − 1.83·21-s + 5.01·22-s − 3.65·23-s + 16.4·24-s − 4.77·25-s + 1.90·26-s + 5.08·27-s + 5.61·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 1.02·3-s + 2.70·4-s + 0.210·5-s + 1.96·6-s + 0.392·7-s − 3.28·8-s + 0.0411·9-s − 0.404·10-s − 0.555·11-s − 2.75·12-s − 0.194·13-s − 0.755·14-s − 0.214·15-s + 3.61·16-s + 0.192·17-s − 0.0792·18-s + 0.839·19-s + 0.568·20-s − 0.400·21-s + 1.06·22-s − 0.761·23-s + 3.34·24-s − 0.955·25-s + 0.374·26-s + 0.978·27-s + 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 547 | \( 1 + T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 - 0.469T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 + 0.700T + 13T^{2} \) |
| 17 | \( 1 - 0.793T + 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 5.52T + 29T^{2} \) |
| 31 | \( 1 - 6.13T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 9.69T + 41T^{2} \) |
| 43 | \( 1 + 4.25T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 + 7.01T + 53T^{2} \) |
| 59 | \( 1 - 6.98T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 + 5.10T + 67T^{2} \) |
| 71 | \( 1 - 1.33T + 71T^{2} \) |
| 73 | \( 1 + 7.83T + 73T^{2} \) |
| 79 | \( 1 - 4.36T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 0.208T + 89T^{2} \) |
| 97 | \( 1 - 3.08T + 97T^{2} \) |
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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
−10.15375292426754281109691091397, −9.775293799943445489551733263635, −8.462572226034336245936665770895, −7.957444403180484682167740160823, −6.86714877393291162791960352591, −6.08344516614092507177803216195, −5.11360281339546546970207398468, −2.90957299938188625807073647819, −1.50439809214026082281975267437, 0,
1.50439809214026082281975267437, 2.90957299938188625807073647819, 5.11360281339546546970207398468, 6.08344516614092507177803216195, 6.86714877393291162791960352591, 7.957444403180484682167740160823, 8.462572226034336245936665770895, 9.775293799943445489551733263635, 10.15375292426754281109691091397
