| L(s) = 1 | + 208.·3-s − 1.44e3·5-s − 8.05e3·7-s + 2.39e4·9-s + 7.41e4·11-s − 2.85e4·13-s − 3.02e5·15-s − 3.78e5·17-s + 8.40e5·19-s − 1.68e6·21-s + 2.29e6·23-s + 1.40e5·25-s + 8.85e5·27-s + 3.59e6·29-s − 4.11e5·31-s + 1.54e7·33-s + 1.16e7·35-s − 1.48e7·37-s − 5.96e6·39-s − 1.38e7·41-s − 3.28e7·43-s − 3.46e7·45-s − 4.00e7·47-s + 2.44e7·49-s − 7.91e7·51-s − 8.39e7·53-s − 1.07e8·55-s + ⋯ |
| L(s) = 1 | + 1.48·3-s − 1.03·5-s − 1.26·7-s + 1.21·9-s + 1.52·11-s − 0.277·13-s − 1.54·15-s − 1.10·17-s + 1.47·19-s − 1.88·21-s + 1.71·23-s + 0.0717·25-s + 0.320·27-s + 0.943·29-s − 0.0799·31-s + 2.27·33-s + 1.31·35-s − 1.30·37-s − 0.412·39-s − 0.763·41-s − 1.46·43-s − 1.25·45-s − 1.19·47-s + 0.606·49-s − 1.63·51-s − 1.46·53-s − 1.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 3 | \( 1 - 208.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.44e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.05e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.41e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.78e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.40e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.29e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.59e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.11e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.48e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.38e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.28e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.00e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.39e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.39e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.87e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.39e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.87e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.67e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.75e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.54e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.17e9T + 7.60e17T^{2} \) |
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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
−9.835002930420432044285170927763, −9.155045206748317586005613405169, −8.460066599533881206166362059240, −7.23122058433312549763818148010, −6.63274038404512049936778908660, −4.61558458993928525711181355470, −3.37004361973436265509490251480, −3.16838297667394621656005310768, −1.47549472449350956807436582882, 0,
1.47549472449350956807436582882, 3.16838297667394621656005310768, 3.37004361973436265509490251480, 4.61558458993928525711181355470, 6.63274038404512049936778908660, 7.23122058433312549763818148010, 8.460066599533881206166362059240, 9.155045206748317586005613405169, 9.835002930420432044285170927763
