| L(s) = 1 | + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + 8.54·5-s + 3.04i·7-s − 7.99·8-s + (8.54 + 14.8i)10-s + 13.0i·11-s − 21.8·13-s + (−5.27 + 3.04i)14-s + (−8 − 13.8i)16-s + 7.27·17-s − 4.35i·19-s + (−17.0 + 29.6i)20-s + (−22.5 + 13.0i)22-s + 31.8i·23-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 1.70·5-s + 0.435i·7-s − 0.999·8-s + (0.854 + 1.48i)10-s + 1.18i·11-s − 1.67·13-s + (−0.376 + 0.217i)14-s + (−0.5 − 0.866i)16-s + 0.427·17-s − 0.229i·19-s + (−0.854 + 1.48i)20-s + (−1.02 + 0.591i)22-s + 1.38i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.525827848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.525827848\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 - 8.54T + 25T^{2} \) |
| 7 | \( 1 - 3.04iT - 49T^{2} \) |
| 11 | \( 1 - 13.0iT - 121T^{2} \) |
| 13 | \( 1 + 21.8T + 169T^{2} \) |
| 17 | \( 1 - 7.27T + 289T^{2} \) |
| 23 | \( 1 - 31.8iT - 529T^{2} \) |
| 29 | \( 1 + 4.37T + 841T^{2} \) |
| 31 | \( 1 - 21.0iT - 961T^{2} \) |
| 37 | \( 1 - 24.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 6.74T + 1.68e3T^{2} \) |
| 43 | \( 1 - 14.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 59.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 26.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 76.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 31.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 25.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 110.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 104. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 0.376iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 47.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 93.6T + 9.40e3T^{2} \) |
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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
−10.26256024100155103580706971468, −9.523608470824817653532408939418, −9.179269949557848758176932921814, −7.71679659768502757306236894125, −7.04341663545547525725686426309, −6.06422327124126257488914404153, −5.27967446528227925270722538712, −4.66692326367219900741325861307, −2.92644962434769505563874399702, −1.94609867323075399952647902889,
0.73218352613788624972160787084, 2.14085629737777073224648067323, 2.89797425856664528825973240516, 4.34617254332896330739044155155, 5.41136006719883589115554428117, 5.95693612747573053155497732611, 7.02119856117028603434874896091, 8.535155262354102836403935961579, 9.396526678589020140777204668559, 10.16886912523226977048334230145
