L(s) = 1 | + (−1 − 1.73i)3-s + (−0.5 − 0.866i)5-s − 4·7-s + (−0.499 + 0.866i)9-s − 11-s + (2 − 3.46i)13-s + (−0.999 + 1.73i)15-s + (−2 − 3.46i)17-s + (−3.5 + 2.59i)19-s + (4 + 6.92i)21-s + (3 − 5.19i)23-s + (−0.499 + 0.866i)25-s − 4.00·27-s + (−2.5 + 4.33i)29-s + 31-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.999i)3-s + (−0.223 − 0.387i)5-s − 1.51·7-s + (−0.166 + 0.288i)9-s − 0.301·11-s + (0.554 − 0.960i)13-s + (−0.258 + 0.447i)15-s + (−0.485 − 0.840i)17-s + (−0.802 + 0.596i)19-s + (0.872 + 1.51i)21-s + (0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s − 0.769·27-s + (−0.464 + 0.804i)29-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (3.5 - 2.59i)T \) |
good | 3 | \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5 - 8.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.5 + 11.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.861817256456531708177074550795, −7.912008019633663019795855793100, −7.11604849509614428722895249577, −6.35779675985696740922609640070, −5.89929474360026346010270530637, −4.75308215230347394487865147496, −3.54070746346573287237937288943, −2.60487759655148434030595759890, −1.02783145871320189576730697162, 0,
2.21538974500818549052233388950, 3.57104816470833818315872683285, 4.00517279255826453743088254989, 5.09008931674576238279297334464, 6.11183702704466892374436785412, 6.60733314705082890815393328489, 7.54933103379988826327758389596, 8.810807804624192130826012710379, 9.392813459486385073392930453541