| L(s) = 1 | + 3-s + 3.19·5-s + 7-s + 9-s − 5·11-s − 2.19·13-s + 3.19·15-s + 17-s − 7.38·19-s + 21-s − 23-s + 5.19·25-s + 27-s − 5·29-s − 7.38·31-s − 5·33-s + 3.19·35-s − 5.38·37-s − 2.19·39-s − 41-s + 0.192·43-s + 3.19·45-s + 0.385·47-s + 49-s + 51-s + 5.57·53-s − 15.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.42·5-s + 0.377·7-s + 0.333·9-s − 1.50·11-s − 0.608·13-s + 0.824·15-s + 0.242·17-s − 1.69·19-s + 0.218·21-s − 0.208·23-s + 1.03·25-s + 0.192·27-s − 0.928·29-s − 1.32·31-s − 0.870·33-s + 0.539·35-s − 0.885·37-s − 0.351·39-s − 0.156·41-s + 0.0293·43-s + 0.475·45-s + 0.0561·47-s + 0.142·49-s + 0.140·51-s + 0.766·53-s − 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.19T + 5T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 7.38T + 19T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 7.38T + 31T^{2} \) |
| 37 | \( 1 + 5.38T + 37T^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 - 0.192T + 43T^{2} \) |
| 47 | \( 1 - 0.385T + 47T^{2} \) |
| 53 | \( 1 - 5.57T + 53T^{2} \) |
| 59 | \( 1 + 3.57T + 59T^{2} \) |
| 61 | \( 1 - 3.80T + 61T^{2} \) |
| 67 | \( 1 - 3.19T + 67T^{2} \) |
| 71 | \( 1 - 2.57T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 0.614T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 3.77T + 97T^{2} \) |
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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
−7.48988892228199219276017745353, −6.95917894497327301570283606295, −5.94601327157683506046132700998, −5.47664138694075781671457427360, −4.82928599466992678616499789068, −3.90937607811772086584339063382, −2.83044343101377629766332249768, −2.17530800916773690356540313391, −1.71649162824622335529075221087, 0,
1.71649162824622335529075221087, 2.17530800916773690356540313391, 2.83044343101377629766332249768, 3.90937607811772086584339063382, 4.82928599466992678616499789068, 5.47664138694075781671457427360, 5.94601327157683506046132700998, 6.95917894497327301570283606295, 7.48988892228199219276017745353
