| L(s) = 1 | − 1.56·3-s + 5-s − 0.438·7-s − 0.561·9-s − 11-s − 7.12·13-s − 1.56·15-s + 4.68·17-s + 5.56·19-s + 0.684·21-s + 7.12·23-s + 25-s + 5.56·27-s − 4.43·29-s + 5.56·31-s + 1.56·33-s − 0.438·35-s − 11.5·37-s + 11.1·39-s + 4.24·41-s + 5.12·43-s − 0.561·45-s − 13.3·47-s − 6.80·49-s − 7.31·51-s + 2.68·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.447·5-s − 0.165·7-s − 0.187·9-s − 0.301·11-s − 1.97·13-s − 0.403·15-s + 1.13·17-s + 1.27·19-s + 0.149·21-s + 1.48·23-s + 0.200·25-s + 1.07·27-s − 0.824·29-s + 0.998·31-s + 0.271·33-s − 0.0741·35-s − 1.90·37-s + 1.78·39-s + 0.663·41-s + 0.781·43-s − 0.0837·45-s − 1.95·47-s − 0.972·49-s − 1.02·51-s + 0.368·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 + 0.438T + 7T^{2} \) |
| 13 | \( 1 + 7.12T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 - 5.56T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 - 8.43T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 + 7.12T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.081849035586042927734999516818, −7.28140531536497405301610553488, −6.79005615202367073744992693170, −5.71789778929994399575546332361, −5.21806970688879035756218153459, −4.79290748157005225642075227038, −3.26261732832436156828400407374, −2.65034276651011634087465381970, −1.26279422969758656554803887559, 0,
1.26279422969758656554803887559, 2.65034276651011634087465381970, 3.26261732832436156828400407374, 4.79290748157005225642075227038, 5.21806970688879035756218153459, 5.71789778929994399575546332361, 6.79005615202367073744992693170, 7.28140531536497405301610553488, 8.081849035586042927734999516818
