| L(s) = 1 | + (0.615 − 1.61i)3-s − 1.31·5-s + (−2.24 − 1.99i)9-s − 4.97i·11-s − 0.733i·13-s + (−0.806 + 2.12i)15-s − 0.728·17-s + 6.94i·19-s − 7.92i·23-s − 3.27·25-s + (−4.60 + 2.40i)27-s + 4.62i·29-s + 1.80i·31-s + (−8.06 − 3.06i)33-s − 5.22·37-s + ⋯ |
| L(s) = 1 | + (0.355 − 0.934i)3-s − 0.586·5-s + (−0.747 − 0.663i)9-s − 1.50i·11-s − 0.203i·13-s + (−0.208 + 0.548i)15-s − 0.176·17-s + 1.59i·19-s − 1.65i·23-s − 0.655·25-s + (−0.886 + 0.463i)27-s + 0.859i·29-s + 0.323i·31-s + (−1.40 − 0.533i)33-s − 0.858·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8517041668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8517041668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.615 + 1.61i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.31T + 5T^{2} \) |
| 11 | \( 1 + 4.97iT - 11T^{2} \) |
| 13 | \( 1 + 0.733iT - 13T^{2} \) |
| 17 | \( 1 + 0.728T + 17T^{2} \) |
| 19 | \( 1 - 6.94iT - 19T^{2} \) |
| 23 | \( 1 + 7.92iT - 23T^{2} \) |
| 29 | \( 1 - 4.62iT - 29T^{2} \) |
| 31 | \( 1 - 1.80iT - 31T^{2} \) |
| 37 | \( 1 + 5.22T + 37T^{2} \) |
| 41 | \( 1 + 7.83T + 41T^{2} \) |
| 43 | \( 1 + 1.27T + 43T^{2} \) |
| 47 | \( 1 - 4.24T + 47T^{2} \) |
| 53 | \( 1 + 14.4iT - 53T^{2} \) |
| 59 | \( 1 + 9.35T + 59T^{2} \) |
| 61 | \( 1 + 13.5iT - 61T^{2} \) |
| 67 | \( 1 + 7.54T + 67T^{2} \) |
| 71 | \( 1 - 6.92iT - 71T^{2} \) |
| 73 | \( 1 + 3.79iT - 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 6.58T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 - 15.1iT - 97T^{2} \) |
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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
−9.066237488409525245698081630885, −8.236034782665887829196366053306, −8.059435295863449363307256820045, −6.83969394231518902927858978346, −6.19623090030550302682596940209, −5.25749345646440325195118977293, −3.76187770450890385895561990401, −3.13765405711439217886461141758, −1.76232711749536285644010102235, −0.33605572291220886317302757033,
2.01689062533602828087917356472, 3.17220608839286449029453435962, 4.23698101163369382709465742178, 4.74205147709809454928662015193, 5.79989324129501985896973743939, 7.17045749414127952261623965755, 7.62190363309006829363616841821, 8.740077167765932219346804188306, 9.410729221348468505243136386122, 10.00648212792720686940884087661
