| L(s) = 1 | + 11-s + 2·13-s − 4·17-s − 2·19-s − 5·23-s − 29-s − 2·31-s − 3·37-s + 12·41-s + 11·43-s + 2·47-s + 6·53-s + 10·59-s − 4·61-s + 67-s − 3·71-s + 9·79-s − 2·83-s − 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s + 0.554·13-s − 0.970·17-s − 0.458·19-s − 1.04·23-s − 0.185·29-s − 0.359·31-s − 0.493·37-s + 1.87·41-s + 1.67·43-s + 0.291·47-s + 0.824·53-s + 1.30·59-s − 0.512·61-s + 0.122·67-s − 0.356·71-s + 1.01·79-s − 0.219·83-s − 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.281943256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.281943256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.04227766812131, −12.84577778945062, −12.20556476943676, −11.81518335116007, −11.20536314750658, −10.83589218625061, −10.49523241429925, −9.790564254312049, −9.366373194863944, −8.768969981537261, −8.604415404554080, −7.816606215059898, −7.440714395495029, −6.872675013363327, −6.287357987841028, −5.906013162028907, −5.450022300728397, −4.635976137198979, −4.122706929082635, −3.872825250460394, −3.093067194911318, −2.270658203546064, −2.068552335193086, −1.110369150568362, −0.4697916307123385,
0.4697916307123385, 1.110369150568362, 2.068552335193086, 2.270658203546064, 3.093067194911318, 3.872825250460394, 4.122706929082635, 4.635976137198979, 5.450022300728397, 5.906013162028907, 6.287357987841028, 6.872675013363327, 7.440714395495029, 7.816606215059898, 8.604415404554080, 8.768969981537261, 9.366373194863944, 9.790564254312049, 10.49523241429925, 10.83589218625061, 11.20536314750658, 11.81518335116007, 12.20556476943676, 12.84577778945062, 13.04227766812131
