| L(s) = 1 | − 1.79·3-s + 3.79·5-s + 4.79·7-s + 0.208·9-s − 11-s + 1.20·13-s − 6.79·15-s − 7.58·17-s − 19-s − 8.58·21-s + 1.58·23-s + 9.37·25-s + 5.00·27-s + 2.20·29-s + 10.7·31-s + 1.79·33-s + 18.1·35-s + 8·37-s − 2.16·39-s − 0.791·41-s − 7.37·43-s + 0.791·45-s − 9.16·47-s + 15.9·49-s + 13.5·51-s − 1.58·53-s − 3.79·55-s + ⋯ |
| L(s) = 1 | − 1.03·3-s + 1.69·5-s + 1.81·7-s + 0.0695·9-s − 0.301·11-s + 0.335·13-s − 1.75·15-s − 1.83·17-s − 0.229·19-s − 1.87·21-s + 0.329·23-s + 1.87·25-s + 0.962·27-s + 0.410·29-s + 1.93·31-s + 0.311·33-s + 3.07·35-s + 1.31·37-s − 0.346·39-s − 0.123·41-s − 1.12·43-s + 0.117·45-s − 1.33·47-s + 2.27·49-s + 1.90·51-s − 0.217·53-s − 0.511·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.188325520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.188325520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 - 3.79T + 5T^{2} \) |
| 7 | \( 1 - 4.79T + 7T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 0.791T + 41T^{2} \) |
| 43 | \( 1 + 7.37T + 43T^{2} \) |
| 47 | \( 1 + 9.16T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4.79T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 9.58T + 73T^{2} \) |
| 79 | \( 1 - 5.58T + 79T^{2} \) |
| 83 | \( 1 - 3.79T + 83T^{2} \) |
| 89 | \( 1 + 4.41T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.486807185866756808031225617644, −8.142490351811044314131412856339, −6.61833617184988910411403824790, −6.46654781869563031625894102157, −5.49619087486239944381125007344, −4.92906925361413416353411607223, −4.48925273948586296338615918432, −2.63289081188703848382747190153, −1.94377543123514766152516822837, −0.983137180192912163317366735551,
0.983137180192912163317366735551, 1.94377543123514766152516822837, 2.63289081188703848382747190153, 4.48925273948586296338615918432, 4.92906925361413416353411607223, 5.49619087486239944381125007344, 6.46654781869563031625894102157, 6.61833617184988910411403824790, 8.142490351811044314131412856339, 8.486807185866756808031225617644
