| L(s) = 1 | + 4·4-s + 9·9-s − 1.31·11-s + 12.0·13-s + 16·16-s − 35.3·19-s + 25·25-s + 56.8·29-s + 36·36-s − 5.24·44-s − 85.4·47-s + 49·49-s + 48.1·52-s − 45.3·53-s + 64·64-s + 98.9·67-s + 138.·73-s − 141.·76-s + 81·81-s − 138.·97-s − 11.8·99-s + 100·100-s − 31.3·113-s + 227.·116-s + 108.·117-s + ⋯ |
| L(s) = 1 | + 4-s + 9-s − 0.119·11-s + 0.925·13-s + 16-s − 1.86·19-s + 25-s + 1.95·29-s + 36-s − 0.119·44-s − 1.81·47-s + 0.999·49-s + 0.925·52-s − 0.856·53-s + 64-s + 1.47·67-s + 1.90·73-s − 1.86·76-s + 81-s − 1.42·97-s − 0.119·99-s + 100-s − 0.277·113-s + 1.95·116-s + 0.925·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.520646248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520646248\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 + 547T \) |
| good | 2 | \( 1 - 4T^{2} \) |
| 3 | \( 1 - 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 1.31T + 121T^{2} \) |
| 13 | \( 1 - 12.0T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 35.3T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 56.8T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 + 85.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 45.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 98.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 138.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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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
−10.64192921059931043260682585954, −9.990110729492097668359572250469, −8.659337806684879330641757127344, −7.931137464318274405672144419938, −6.62741824993304671830727285447, −6.46471886661242904474601187143, −4.92282029190601306714769922749, −3.78973971150953595697167912247, −2.48773918926333368846680989935, −1.26184553004121869175853389375,
1.26184553004121869175853389375, 2.48773918926333368846680989935, 3.78973971150953595697167912247, 4.92282029190601306714769922749, 6.46471886661242904474601187143, 6.62741824993304671830727285447, 7.931137464318274405672144419938, 8.659337806684879330641757127344, 9.990110729492097668359572250469, 10.64192921059931043260682585954
