L(s) = 1 | + (1.61 − 0.618i)3-s − 1.23i·5-s − 3.23i·7-s + (2.23 − 2.00i)9-s − 0.763·11-s − 4.47·13-s + (−0.763 − 2.00i)15-s + 6.47i·17-s − 5.23i·19-s + (−2.00 − 5.23i)21-s + 6.47·23-s + 3.47·25-s + (2.38 − 4.61i)27-s + 9.23i·29-s + 0.763i·31-s + ⋯ |
L(s) = 1 | + (0.934 − 0.356i)3-s − 0.552i·5-s − 1.22i·7-s + (0.745 − 0.666i)9-s − 0.230·11-s − 1.24·13-s + (−0.197 − 0.516i)15-s + 1.56i·17-s − 1.20i·19-s + (−0.436 − 1.14i)21-s + 1.34·23-s + 0.694·25-s + (0.458 − 0.888i)27-s + 1.71i·29-s + 0.137i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45044 - 0.998629i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45044 - 0.998629i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.61 + 0.618i)T \) |
good | 5 | \( 1 + 1.23iT - 5T^{2} \) |
| 7 | \( 1 + 3.23iT - 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 6.47iT - 17T^{2} \) |
| 19 | \( 1 + 5.23iT - 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 - 9.23iT - 29T^{2} \) |
| 31 | \( 1 - 0.763iT - 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 + 2.47iT - 41T^{2} \) |
| 43 | \( 1 - 2.76iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 1.23iT - 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 - 3.70iT - 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 13.7iT - 79T^{2} \) |
| 83 | \( 1 + 7.23T + 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−10.96232619644934587218618543539, −10.22985321993568977451284944312, −9.155500414776418872618736695583, −8.459874022653092865895918838826, −7.31167834096556364237367427665, −6.87608787923098798869876049769, −5.06366884368233726996521205517, −4.08716589593396748958705805894, −2.81637949391504147846896050728, −1.18538501525557478656219182913,
2.38816110549454468176908182544, 2.99189521466023003254220494210, 4.59214357569285728443681311873, 5.58033037672935558922287635504, 7.04115419664090146379155592869, 7.81154220456290406141470382148, 8.913943113535815542633723657084, 9.577820954804888068361625544663, 10.37638248633915746304477388016, 11.57931989203260499114659443858