| L(s) = 1 | − 0.405i·3-s + 1.20i·5-s − 0.122i·7-s + 2.83·9-s + 1.65i·11-s − 1.21·13-s + 0.488·15-s + (−4.09 + 0.446i)17-s + 5.49·19-s − 0.0496·21-s − 1.29i·23-s + 3.54·25-s − 2.36i·27-s − 9.31i·29-s − 6.06i·31-s + ⋯ |
| L(s) = 1 | − 0.233i·3-s + 0.538i·5-s − 0.0462i·7-s + 0.945·9-s + 0.498i·11-s − 0.337·13-s + 0.126·15-s + (−0.994 + 0.108i)17-s + 1.26·19-s − 0.0108·21-s − 0.269i·23-s + 0.709·25-s − 0.454i·27-s − 1.72i·29-s − 1.08i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.063342975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.063342975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (4.09 - 0.446i)T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 0.405iT - 3T^{2} \) |
| 5 | \( 1 - 1.20iT - 5T^{2} \) |
| 7 | \( 1 + 0.122iT - 7T^{2} \) |
| 11 | \( 1 - 1.65iT - 11T^{2} \) |
| 13 | \( 1 + 1.21T + 13T^{2} \) |
| 19 | \( 1 - 5.49T + 19T^{2} \) |
| 23 | \( 1 + 1.29iT - 23T^{2} \) |
| 29 | \( 1 + 9.31iT - 29T^{2} \) |
| 31 | \( 1 + 6.06iT - 31T^{2} \) |
| 37 | \( 1 - 4.07iT - 37T^{2} \) |
| 41 | \( 1 - 1.98iT - 41T^{2} \) |
| 43 | \( 1 - 7.44T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 61 | \( 1 - 3.86iT - 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 - 12.5iT - 71T^{2} \) |
| 73 | \( 1 - 14.7iT - 73T^{2} \) |
| 79 | \( 1 + 0.917iT - 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 2.24T + 89T^{2} \) |
| 97 | \( 1 + 7.92iT - 97T^{2} \) |
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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
−8.310527983973165346602173296185, −7.58681994783303098253091732845, −7.02227648630797534247322642798, −6.45540835181264792107260901428, −5.53031230478764788026741578549, −4.52693977025803491650515308216, −4.02161268257019146879878187593, −2.79398876239553516663910194757, −2.09665753356158987836180234228, −0.854785282133091313732816738170,
0.841086007343673728656083352250, 1.82368785904685883874723491503, 3.07650724139413755186994393918, 3.81956612161386746582646232450, 4.94756991084700169980325135003, 5.07698757126377604428298127494, 6.27012217514218614859013690239, 7.09321705918061736925479074617, 7.56544583909486396342494426710, 8.659862785577928638021377055377
