L(s) = 1 | + (−0.212 + 0.154i)2-s + (−0.596 + 1.83i)4-s + (0.809 + 0.587i)5-s + (−0.986 + 3.03i)7-s + (−0.318 − 0.980i)8-s − 0.262·10-s + (−3.27 + 0.547i)11-s + (0.905 − 0.658i)13-s + (−0.258 − 0.796i)14-s + (−2.90 − 2.11i)16-s + (−0.0713 − 0.0518i)17-s + (−0.0212 − 0.0654i)19-s + (−1.56 + 1.13i)20-s + (0.609 − 0.620i)22-s − 6.65·23-s + ⋯ |
L(s) = 1 | + (−0.150 + 0.109i)2-s + (−0.298 + 0.918i)4-s + (0.361 + 0.262i)5-s + (−0.372 + 1.14i)7-s + (−0.112 − 0.346i)8-s − 0.0829·10-s + (−0.986 + 0.165i)11-s + (0.251 − 0.182i)13-s + (−0.0691 − 0.212i)14-s + (−0.726 − 0.527i)16-s + (−0.0173 − 0.0125i)17-s + (−0.00487 − 0.0150i)19-s + (−0.349 + 0.253i)20-s + (0.130 − 0.132i)22-s − 1.38·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.199409 + 0.803866i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.199409 + 0.803866i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (3.27 - 0.547i)T \) |
good | 2 | \( 1 + (0.212 - 0.154i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (0.986 - 3.03i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.905 + 0.658i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.0713 + 0.0518i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.0212 + 0.0654i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 6.65T + 23T^{2} \) |
| 29 | \( 1 + (1.15 - 3.55i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-7.75 + 5.63i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.57 - 7.92i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.60 - 11.0i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + (0.280 + 0.863i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.705 + 0.512i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.567 - 1.74i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-8.13 - 5.91i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 9.53T + 67T^{2} \) |
| 71 | \( 1 + (3.77 + 2.74i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.21 - 6.80i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.640 - 0.465i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.200 - 0.145i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + (-3.31 + 2.40i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.55306126810301534388957622655, −10.17131297896532015294759711555, −9.581318182426259192483357475953, −8.438902487548763121777086938612, −8.003683107281503484129246633823, −6.72693575657578000948686893251, −5.81531363194089514054447035679, −4.69105592222094969319408571844, −3.25781397585747330690609083637, −2.38518640892203230353886079196,
0.50394352307388378666981602917, 2.06462993139650954837751504465, 3.79226459359172011655807744786, 4.90304786865758778521973105232, 5.85221819089097321745086929482, 6.78500778785077872037979844693, 7.955824856844631832503432613821, 8.895359134160342540650676377470, 10.07219005365280803312266721151, 10.23227820874030582939625570025