| L(s) = 1 | + 3-s − 3.33·5-s + 7-s + 9-s − 4.27·11-s − 3.33·15-s − 6.05·17-s + 19-s + 21-s − 3.25·23-s + 6.12·25-s + 27-s + 8.45·29-s − 2.27·31-s − 4.27·33-s − 3.33·35-s − 0.724·37-s − 0.127·41-s + 8.99·43-s − 3.33·45-s − 7.47·47-s + 49-s − 6.05·51-s + 2.21·53-s + 14.2·55-s + 57-s − 4.79·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.49·5-s + 0.377·7-s + 0.333·9-s − 1.28·11-s − 0.861·15-s − 1.46·17-s + 0.229·19-s + 0.218·21-s − 0.677·23-s + 1.22·25-s + 0.192·27-s + 1.57·29-s − 0.407·31-s − 0.743·33-s − 0.563·35-s − 0.119·37-s − 0.0198·41-s + 1.37·43-s − 0.497·45-s − 1.09·47-s + 0.142·49-s − 0.848·51-s + 0.303·53-s + 1.92·55-s + 0.132·57-s − 0.624·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.188462782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.188462782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3.33T + 5T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 6.05T + 17T^{2} \) |
| 23 | \( 1 + 3.25T + 23T^{2} \) |
| 29 | \( 1 - 8.45T + 29T^{2} \) |
| 31 | \( 1 + 2.27T + 31T^{2} \) |
| 37 | \( 1 + 0.724T + 37T^{2} \) |
| 41 | \( 1 + 0.127T + 41T^{2} \) |
| 43 | \( 1 - 8.99T + 43T^{2} \) |
| 47 | \( 1 + 7.47T + 47T^{2} \) |
| 53 | \( 1 - 2.21T + 53T^{2} \) |
| 59 | \( 1 + 4.79T + 59T^{2} \) |
| 61 | \( 1 + 6.54T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 1.91T + 71T^{2} \) |
| 73 | \( 1 + 0.127T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 6.08T + 83T^{2} \) |
| 89 | \( 1 - 1.57T + 89T^{2} \) |
| 97 | \( 1 + 4.39T + 97T^{2} \) |
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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
−7.966862553537395311815723947394, −7.57644480010168831202537444242, −6.86018901138345790249066965653, −5.95015230388809409057853846168, −4.72854082392121081048973230455, −4.55077617903629729282761445609, −3.58416505730526529490629768261, −2.83913192923825981371787810362, −2.01103119428951373595939811269, −0.52616893575508350564251437079,
0.52616893575508350564251437079, 2.01103119428951373595939811269, 2.83913192923825981371787810362, 3.58416505730526529490629768261, 4.55077617903629729282761445609, 4.72854082392121081048973230455, 5.95015230388809409057853846168, 6.86018901138345790249066965653, 7.57644480010168831202537444242, 7.966862553537395311815723947394
