| L(s) = 1 | + (1 − 1.73i)2-s + 3-s + (−1.99 − 3.46i)4-s − 5.19i·5-s + (1 − 1.73i)6-s − 7.99·8-s − 8·9-s + (−9 − 5.19i)10-s + 17·11-s + (−1.99 − 3.46i)12-s − 13.8i·13-s − 5.19i·15-s + (−8 + 13.8i)16-s − 25·17-s + (−8 + 13.8i)18-s − 7·19-s + ⋯ |
| L(s) = 1 | + (0.5 − 0.866i)2-s + 0.333·3-s + (−0.499 − 0.866i)4-s − 1.03i·5-s + (0.166 − 0.288i)6-s − 0.999·8-s − 0.888·9-s + (−0.900 − 0.519i)10-s + 1.54·11-s + (−0.166 − 0.288i)12-s − 1.06i·13-s − 0.346i·15-s + (−0.5 + 0.866i)16-s − 1.47·17-s + (−0.444 + 0.769i)18-s − 0.368·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.77248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77248i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + 9T^{2} \) |
| 5 | \( 1 + 5.19iT - 25T^{2} \) |
| 11 | \( 1 - 17T + 121T^{2} \) |
| 13 | \( 1 + 13.8iT - 169T^{2} \) |
| 17 | \( 1 + 25T + 289T^{2} \) |
| 19 | \( 1 + 7T + 361T^{2} \) |
| 23 | \( 1 - 5.19iT - 529T^{2} \) |
| 29 | \( 1 - 13.8iT - 841T^{2} \) |
| 31 | \( 1 + 32.9iT - 961T^{2} \) |
| 37 | \( 1 - 8.66iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 26T + 1.68e3T^{2} \) |
| 43 | \( 1 - 14T + 1.84e3T^{2} \) |
| 47 | \( 1 + 50.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 91.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 55T + 3.48e3T^{2} \) |
| 61 | \( 1 - 22.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 17T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 119T + 5.32e3T^{2} \) |
| 79 | \( 1 - 74.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 110T + 6.88e3T^{2} \) |
| 89 | \( 1 - 71T + 7.92e3T^{2} \) |
| 97 | \( 1 + 22T + 9.40e3T^{2} \) |
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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
−10.91241477031188834571312157269, −9.597068933605756012254339930955, −8.940987138991901716005450790750, −8.317678713292489473789154816897, −6.53917125855304984631659421085, −5.53278546043108978719885127940, −4.49592521713073698533991253506, −3.50339332797582734224368031948, −2.10265529572775601168692473659, −0.62489167193046042945901485616,
2.44920280773958853497853200196, 3.65717963462851389988986036839, 4.59027064935585663304125268820, 6.34345987646865853333486274220, 6.49163935087303854866438684708, 7.61133244539580170409930637683, 8.950854831764983805183615387945, 9.144558870772642631725640140693, 10.88415940219253177777257996282, 11.56548568022576604990603617517
