| L(s) = 1 | + 3-s − 2.39i·5-s − 1.09·7-s + 9-s + (2.27 + 2.41i)11-s − 3.59·13-s − 2.39i·15-s − 5.49i·17-s − 7.68i·19-s − 1.09·21-s + 3.06i·23-s − 0.756·25-s + 27-s − 8.48·29-s + 0.243i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.07i·5-s − 0.414·7-s + 0.333·9-s + (0.686 + 0.726i)11-s − 0.996·13-s − 0.619i·15-s − 1.33i·17-s − 1.76i·19-s − 0.239·21-s + 0.639i·23-s − 0.151·25-s + 0.192·27-s − 1.57·29-s + 0.0437i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.500839419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500839419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + (-2.27 - 2.41i)T \) |
| good | 5 | \( 1 + 2.39iT - 5T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 13 | \( 1 + 3.59T + 13T^{2} \) |
| 17 | \( 1 + 5.49iT - 17T^{2} \) |
| 19 | \( 1 + 7.68iT - 19T^{2} \) |
| 23 | \( 1 - 3.06iT - 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 - 0.243iT - 31T^{2} \) |
| 37 | \( 1 + 3.09iT - 37T^{2} \) |
| 41 | \( 1 - 6.51iT - 41T^{2} \) |
| 43 | \( 1 + 3.29iT - 43T^{2} \) |
| 47 | \( 1 + 6.95iT - 47T^{2} \) |
| 53 | \( 1 + 10.7iT - 53T^{2} \) |
| 59 | \( 1 + 5.88T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 6.15iT - 71T^{2} \) |
| 73 | \( 1 - 5.98iT - 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 7.18iT - 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−9.082112300154449194819673815249, −8.149905515627108004372155145230, −7.11757295534705799759460344074, −6.89641656231006232086683938171, −5.31305298667918247723661011999, −4.87571439688993982405718209064, −3.96337111853902930289920270652, −2.85064630438315993686086337560, −1.86832274611943964403603775505, −0.46863513131941846632020402715,
1.62245855572617239872207601986, 2.69720145323862677588816202363, 3.55394596942890085801312219799, 4.13159674287174109083535238737, 5.63240548504917212125657093197, 6.29743817678006456408360024756, 7.02123362795878531919929310556, 7.85122089035764027545637470072, 8.489274500425042057023678148432, 9.470253051162879380786143899419
