L(s) = 1 | + 3.60·5-s − 3.60·7-s − 11-s + 4·13-s + 7.21·19-s + 6·23-s + 7.99·25-s − 7.21·29-s + 3.60·31-s − 12.9·35-s + 10·37-s − 7.21·41-s − 7.21·43-s + 10·47-s + 5.99·49-s − 3.60·53-s − 3.60·55-s + 4·59-s + 14.4·65-s − 7.21·67-s − 8·71-s − 3·73-s + 3.60·77-s + 14.4·79-s − 9·83-s + 7.21·89-s − 14.4·91-s + ⋯ |
L(s) = 1 | + 1.61·5-s − 1.36·7-s − 0.301·11-s + 1.10·13-s + 1.65·19-s + 1.25·23-s + 1.59·25-s − 1.33·29-s + 0.647·31-s − 2.19·35-s + 1.64·37-s − 1.12·41-s − 1.09·43-s + 1.45·47-s + 0.857·49-s − 0.495·53-s − 0.486·55-s + 0.520·59-s + 1.78·65-s − 0.880·67-s − 0.949·71-s − 0.351·73-s + 0.410·77-s + 1.62·79-s − 0.987·83-s + 0.764·89-s − 1.51·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.919540292\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.919540292\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.60T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 + 3.60T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 7.21T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 - 7.21T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−9.964840343938564478644416184463, −9.452113455488338039701609058853, −8.830254827021835108430879729823, −7.44872784047950347991868238866, −6.49335744037121874476973366301, −5.89942799257367761273399751420, −5.12993293925237691320274525243, −3.49977165664856287199112412591, −2.68604938373979088836071886122, −1.23059609338931193515469368251,
1.23059609338931193515469368251, 2.68604938373979088836071886122, 3.49977165664856287199112412591, 5.12993293925237691320274525243, 5.89942799257367761273399751420, 6.49335744037121874476973366301, 7.44872784047950347991868238866, 8.830254827021835108430879729823, 9.452113455488338039701609058853, 9.964840343938564478644416184463