| L(s) = 1 | − i·3-s + (−1.41 + 1.73i)5-s − 9-s − 3.46i·13-s + (1.73 + 1.41i)15-s + 4.89i·17-s + 4.89·19-s + 2.82i·23-s + (−0.999 − 4.89i)25-s + i·27-s − 2.82·29-s − 6.92·31-s − 3.46i·37-s − 3.46·39-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.632 + 0.774i)5-s − 0.333·9-s − 0.960i·13-s + (0.447 + 0.365i)15-s + 1.18i·17-s + 1.12·19-s + 0.589i·23-s + (−0.199 − 0.979i)25-s + 0.192i·27-s − 0.525·29-s − 1.24·31-s − 0.569i·37-s − 0.554·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4173479535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4173479535\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.41 - 1.73i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 3.46iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 9.79iT - 97T^{2} \) |
| 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
−8.583835028147087586679542589741, −7.79793226866191452926187717420, −7.47187390279406393136364041194, −6.67938766064439346220324501841, −5.83631454829458159932446498594, −5.24085915489961902226617255967, −3.89033139289274903189320406055, −3.38728611577011254683751093070, −2.43851389725414668147135616310, −1.29005928451778764496440675244,
0.12778054620283715868307987430, 1.44500340757451662509829054457, 2.74316500166825353903046216529, 3.71730314021554491625331751356, 4.35898773672299532696019690962, 5.14708615682268747324669734483, 5.67134374819641275497293972036, 7.01407659171277437033444112452, 7.33833808803672211632950029034, 8.408998057315711526715199354977
