| L(s) = 1 | + 2.06·3-s + 0.0661·5-s + 2.71·7-s + 1.27·9-s − 0.964·11-s − 2.42·13-s + 0.136·15-s + 17-s − 1.56·19-s + 5.60·21-s − 1.66·23-s − 4.99·25-s − 3.57·27-s − 9.05·29-s − 2.31·31-s − 1.99·33-s + 0.179·35-s − 2.65·37-s − 5.00·39-s − 1.95·41-s − 5.12·43-s + 0.0840·45-s − 8.90·47-s + 0.345·49-s + 2.06·51-s − 3.09·53-s − 0.0637·55-s + ⋯ |
| L(s) = 1 | + 1.19·3-s + 0.0295·5-s + 1.02·7-s + 0.423·9-s − 0.290·11-s − 0.671·13-s + 0.0352·15-s + 0.242·17-s − 0.359·19-s + 1.22·21-s − 0.346·23-s − 0.999·25-s − 0.687·27-s − 1.68·29-s − 0.415·31-s − 0.347·33-s + 0.0302·35-s − 0.436·37-s − 0.801·39-s − 0.304·41-s − 0.781·43-s + 0.0125·45-s − 1.29·47-s + 0.0493·49-s + 0.289·51-s − 0.425·53-s − 0.00859·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.06T + 3T^{2} \) |
| 5 | \( 1 - 0.0661T + 5T^{2} \) |
| 7 | \( 1 - 2.71T + 7T^{2} \) |
| 11 | \( 1 + 0.964T + 11T^{2} \) |
| 13 | \( 1 + 2.42T + 13T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 + 9.05T + 29T^{2} \) |
| 31 | \( 1 + 2.31T + 31T^{2} \) |
| 37 | \( 1 + 2.65T + 37T^{2} \) |
| 41 | \( 1 + 1.95T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 + 8.90T + 47T^{2} \) |
| 53 | \( 1 + 3.09T + 53T^{2} \) |
| 61 | \( 1 - 4.52T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 2.32T + 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 + 1.08T + 79T^{2} \) |
| 83 | \( 1 - 6.42T + 83T^{2} \) |
| 89 | \( 1 + 8.02T + 89T^{2} \) |
| 97 | \( 1 - 3.86T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66515632913981936243785523012, −7.12941860491229690654163485652, −6.02993610458138057174110861554, −5.31096807855275061984933245181, −4.61526360503953071876629756093, −3.75114870577199889709716839673, −3.12800922415786553610624616452, −2.03259118962573640868960114684, −1.79081236798950943580421182870, 0,
1.79081236798950943580421182870, 2.03259118962573640868960114684, 3.12800922415786553610624616452, 3.75114870577199889709716839673, 4.61526360503953071876629756093, 5.31096807855275061984933245181, 6.02993610458138057174110861554, 7.12941860491229690654163485652, 7.66515632913981936243785523012
