| L(s) = 1 | + 0.860·3-s + 2.46·5-s − 7-s − 2.25·9-s − 3.72·11-s − 4.64·13-s + 2.11·15-s − 17-s + 0.925·19-s − 0.860·21-s + 6.64·23-s + 1.06·25-s − 4.52·27-s + 4.92·29-s + 5.90·31-s − 3.20·33-s − 2.46·35-s + 8.92·37-s − 4·39-s − 3.93·41-s + 9.50·43-s − 5.56·45-s − 3.44·47-s + 49-s − 0.860·51-s + 4.33·53-s − 9.16·55-s + ⋯ |
| L(s) = 1 | + 0.496·3-s + 1.10·5-s − 0.377·7-s − 0.753·9-s − 1.12·11-s − 1.28·13-s + 0.547·15-s − 0.242·17-s + 0.212·19-s − 0.187·21-s + 1.38·23-s + 0.212·25-s − 0.871·27-s + 0.914·29-s + 1.06·31-s − 0.557·33-s − 0.416·35-s + 1.46·37-s − 0.640·39-s − 0.614·41-s + 1.44·43-s − 0.829·45-s − 0.502·47-s + 0.142·49-s − 0.120·51-s + 0.595·53-s − 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.199561440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.199561440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.860T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 19 | \( 1 - 0.925T + 19T^{2} \) |
| 23 | \( 1 - 6.64T + 23T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 - 5.90T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 + 3.93T + 41T^{2} \) |
| 43 | \( 1 - 9.50T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 - 4.33T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 - 7.37T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 3.07T + 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.84614337585072361409137465514, −7.28190519504846817900887472272, −6.40033220215857411425824997002, −5.78686367258602013074102082635, −5.10375648585788414404913174204, −4.50804974138129668742029522198, −3.10404342381267458433023239974, −2.69553688523424103159568618463, −2.12988041443074167509022562282, −0.68875027615424886941536215636,
0.68875027615424886941536215636, 2.12988041443074167509022562282, 2.69553688523424103159568618463, 3.10404342381267458433023239974, 4.50804974138129668742029522198, 5.10375648585788414404913174204, 5.78686367258602013074102082635, 6.40033220215857411425824997002, 7.28190519504846817900887472272, 7.84614337585072361409137465514
