L(s) = 1 | + 0.705·3-s + 2.65i·5-s − i·7-s − 2.50·9-s − i·11-s + (3.54 − 0.674i)13-s + 1.87i·15-s − 0.198·17-s − 5.34i·19-s − 0.705i·21-s + 6.13·23-s − 2.05·25-s − 3.88·27-s + 2.90·29-s + 5.55i·31-s + ⋯ |
L(s) = 1 | + 0.407·3-s + 1.18i·5-s − 0.377i·7-s − 0.834·9-s − 0.301i·11-s + (0.982 − 0.187i)13-s + 0.483i·15-s − 0.0481·17-s − 1.22i·19-s − 0.153i·21-s + 1.28·23-s − 0.410·25-s − 0.747·27-s + 0.539·29-s + 0.998i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.144939967\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.144939967\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-3.54 + 0.674i)T \) |
good | 3 | \( 1 - 0.705T + 3T^{2} \) |
| 5 | \( 1 - 2.65iT - 5T^{2} \) |
| 17 | \( 1 + 0.198T + 17T^{2} \) |
| 19 | \( 1 + 5.34iT - 19T^{2} \) |
| 23 | \( 1 - 6.13T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 - 5.55iT - 31T^{2} \) |
| 37 | \( 1 + 3.81iT - 37T^{2} \) |
| 41 | \( 1 - 0.304iT - 41T^{2} \) |
| 43 | \( 1 + 2.09T + 43T^{2} \) |
| 47 | \( 1 - 3.81iT - 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + 14.5iT - 59T^{2} \) |
| 61 | \( 1 - 9.96T + 61T^{2} \) |
| 67 | \( 1 - 6.86iT - 67T^{2} \) |
| 71 | \( 1 + 14.7iT - 71T^{2} \) |
| 73 | \( 1 - 11.4iT - 73T^{2} \) |
| 79 | \( 1 + 4.78T + 79T^{2} \) |
| 83 | \( 1 + 15.0iT - 83T^{2} \) |
| 89 | \( 1 - 7.28iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.617738838754465159639526810002, −7.70290406558088931021351461285, −6.91029398990809491430363997328, −6.48299267439748212626426984275, −5.57292735669921815476730565821, −4.69379464222490335361698285841, −3.44473340059270393317476386855, −3.15292787382139147219953961768, −2.27375458026307832020649236684, −0.790078012668799334727735031853,
0.880266726685103744072712724775, 1.87926183071185865139746704436, 2.94330937066251421445561800518, 3.83193535139042631244196272302, 4.66301445781203648768967393778, 5.51346613524663474011194347872, 5.99916676773707291287880116291, 7.01074029333536720642855135128, 8.007603644011798100270592039708, 8.571041809961875323270430181824