| L(s) = 1 | + 3-s + 4.14·5-s + 2.41·7-s + 9-s − 3.46·11-s + 4.14·15-s − 5.17·17-s + 3.46·19-s + 2.41·21-s + 2·23-s + 12.1·25-s + 27-s + 3.17·29-s + 1.05·31-s − 3.46·33-s + 10·35-s − 7.60·37-s − 0.680·41-s + 12.1·43-s + 4.14·45-s + 10.3·47-s − 1.17·49-s − 5.17·51-s + 1.17·53-s − 14.3·55-s + 3.46·57-s − 11.7·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.85·5-s + 0.911·7-s + 0.333·9-s − 1.04·11-s + 1.07·15-s − 1.25·17-s + 0.794·19-s + 0.526·21-s + 0.417·23-s + 2.43·25-s + 0.192·27-s + 0.590·29-s + 0.188·31-s − 0.603·33-s + 1.69·35-s − 1.25·37-s − 0.106·41-s + 1.85·43-s + 0.617·45-s + 1.51·47-s − 0.168·49-s − 0.725·51-s + 0.161·53-s − 1.93·55-s + 0.458·57-s − 1.53·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.284850419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.284850419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4.14T + 5T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 17 | \( 1 + 5.17T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 - 1.05T + 31T^{2} \) |
| 37 | \( 1 + 7.60T + 37T^{2} \) |
| 41 | \( 1 + 0.680T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 2.41T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 1.82T + 79T^{2} \) |
| 83 | \( 1 + 1.36T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−9.070791919329329629574551712953, −8.597308431476677750370056901130, −7.59609458284390640963826227148, −6.82805801462299800089923369862, −5.83103540168679521977385839844, −5.19363998644716118551765009225, −4.44304909659423165112555811789, −2.86030765156726648319526705433, −2.27393052505035443164563026201, −1.34348932482890564086818935588,
1.34348932482890564086818935588, 2.27393052505035443164563026201, 2.86030765156726648319526705433, 4.44304909659423165112555811789, 5.19363998644716118551765009225, 5.83103540168679521977385839844, 6.82805801462299800089923369862, 7.59609458284390640963826227148, 8.597308431476677750370056901130, 9.070791919329329629574551712953
