| L(s) = 1 | + 3-s + 0.925·5-s − 2.76·7-s + 9-s + 2.15·11-s − 0.980·13-s + 0.925·15-s + 4.96·17-s − 7.23·19-s − 2.76·21-s − 6.49·23-s − 4.14·25-s + 27-s − 0.737·29-s + 0.415·31-s + 2.15·33-s − 2.55·35-s + 11.4·37-s − 0.980·39-s + 5.79·41-s − 6.31·43-s + 0.925·45-s + 12.1·47-s + 0.645·49-s + 4.96·51-s − 6.55·53-s + 1.99·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.413·5-s − 1.04·7-s + 0.333·9-s + 0.648·11-s − 0.271·13-s + 0.239·15-s + 1.20·17-s − 1.66·19-s − 0.603·21-s − 1.35·23-s − 0.828·25-s + 0.192·27-s − 0.137·29-s + 0.0745·31-s + 0.374·33-s − 0.432·35-s + 1.87·37-s − 0.157·39-s + 0.905·41-s − 0.963·43-s + 0.137·45-s + 1.76·47-s + 0.0921·49-s + 0.694·51-s − 0.900·53-s + 0.268·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 0.925T + 5T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 + 0.980T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 + 0.737T + 29T^{2} \) |
| 31 | \( 1 - 0.415T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 + 6.31T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 6.55T + 53T^{2} \) |
| 59 | \( 1 + 3.30T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 8.06T + 79T^{2} \) |
| 83 | \( 1 - 4.37T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 5.79T + 97T^{2} \) |
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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
−7.64165042812217457620043003854, −6.68064659544208623292264485461, −6.12216493561284505461203034576, −5.68010638618486816767008850289, −4.32351267939519035736107931435, −3.98495572675873999379187377484, −3.02360104539158246870156380918, −2.33326938488455236613261176594, −1.40527607006618159850014428401, 0,
1.40527607006618159850014428401, 2.33326938488455236613261176594, 3.02360104539158246870156380918, 3.98495572675873999379187377484, 4.32351267939519035736107931435, 5.68010638618486816767008850289, 6.12216493561284505461203034576, 6.68064659544208623292264485461, 7.64165042812217457620043003854
