| L(s) = 1 | + (2 + 2.23i)3-s + 4i·5-s + 8.94·7-s + (−1.00 + 8.94i)9-s + 4.47i·11-s − 17.8·13-s + (−8.94 + 8i)15-s + 17.8i·17-s − 20·19-s + (17.8 + 20.0i)21-s + 16i·23-s + 9·25-s + (−22.0 + 15.6i)27-s − 52i·29-s + 26.8·31-s + ⋯ |
| L(s) = 1 | + (0.666 + 0.745i)3-s + 0.800i·5-s + 1.27·7-s + (−0.111 + 0.993i)9-s + 0.406i·11-s − 1.37·13-s + (−0.596 + 0.533i)15-s + 1.05i·17-s − 1.05·19-s + (0.851 + 0.952i)21-s + 0.695i·23-s + 0.359·25-s + (−0.814 + 0.579i)27-s − 1.79i·29-s + 0.865·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.218340717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.218340717\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2 - 2.23i)T \) |
| good | 5 | \( 1 - 4iT - 25T^{2} \) |
| 7 | \( 1 - 8.94T + 49T^{2} \) |
| 11 | \( 1 - 4.47iT - 121T^{2} \) |
| 13 | \( 1 + 17.8T + 169T^{2} \) |
| 17 | \( 1 - 17.8iT - 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 - 16iT - 529T^{2} \) |
| 29 | \( 1 + 52iT - 841T^{2} \) |
| 31 | \( 1 - 26.8T + 961T^{2} \) |
| 37 | \( 1 - 53.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 35.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 36T + 1.84e3T^{2} \) |
| 47 | \( 1 + 64iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 102. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 17.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 44T + 4.48e3T^{2} \) |
| 71 | \( 1 - 80iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 50T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 160. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 50T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21866631260951268060139536807, −9.861660391813704617821133218078, −8.601257029150458690669249059606, −7.961935105993894353942349601549, −7.22293457224156147629864660194, −5.94872152760245642028879334974, −4.71566605556065479653980032667, −4.20037261379239731103158069357, −2.76040492837630635077174885076, −1.95519933859168440360796741706,
0.67951573783054747627826667082, 1.88653723055840676281703651905, 2.91933185397365793210650904702, 4.54932955850544896042376321185, 5.06560319357146769857481698208, 6.46650837571113703805804540292, 7.41320204108469337871796826183, 8.148066634036616668849024449996, 8.778299353078485985029104610529, 9.524142610818490697985370720189
