| L(s) = 1 | − 3.64·5-s + 7-s − 0.354·11-s − 4.29·13-s + 3.64·17-s + 4.29·19-s − 0.354·23-s + 8.29·25-s + 3.29·29-s + 9.29·31-s − 3.64·35-s − 5·37-s − 3.29·41-s + 9.29·43-s − 3.64·47-s − 6·49-s − 12·53-s + 1.29·55-s − 14.9·59-s + 8.29·61-s + 15.6·65-s − 7.58·67-s − 15.2·71-s + 5·73-s − 0.354·77-s − 10.2·79-s − 12·83-s + ⋯ |
| L(s) = 1 | − 1.63·5-s + 0.377·7-s − 0.106·11-s − 1.19·13-s + 0.884·17-s + 0.984·19-s − 0.0738·23-s + 1.65·25-s + 0.611·29-s + 1.66·31-s − 0.616·35-s − 0.821·37-s − 0.514·41-s + 1.41·43-s − 0.531·47-s − 0.857·49-s − 1.64·53-s + 0.174·55-s − 1.94·59-s + 1.06·61-s + 1.94·65-s − 0.926·67-s − 1.81·71-s + 0.585·73-s − 0.0403·77-s − 1.15·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 0.354T + 11T^{2} \) |
| 13 | \( 1 + 4.29T + 13T^{2} \) |
| 17 | \( 1 - 3.64T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 + 0.354T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 - 9.29T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 - 9.29T + 43T^{2} \) |
| 47 | \( 1 + 3.64T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 - 8.29T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 2.93T + 89T^{2} \) |
| 97 | \( 1 + 1.70T + 97T^{2} \) |
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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
−8.025065500274080504733551270978, −7.59578361671438742100982367772, −7.03777305914859827849708543611, −5.95227196659194176390266399095, −4.81612751489609415792154340702, −4.56552241887556370295646166041, −3.39325739348560635164175647502, −2.82714185758809143747148933728, −1.26122733187792306600673514828, 0,
1.26122733187792306600673514828, 2.82714185758809143747148933728, 3.39325739348560635164175647502, 4.56552241887556370295646166041, 4.81612751489609415792154340702, 5.95227196659194176390266399095, 7.03777305914859827849708543611, 7.59578361671438742100982367772, 8.025065500274080504733551270978
