L(s) = 1 | + 2.04·2-s − 0.838i·3-s + 2.17·4-s − 5-s − 1.71i·6-s + 4.78i·7-s + 0.358·8-s + 2.29·9-s − 2.04·10-s + (−2.69 + 1.93i)11-s − 1.82i·12-s + 4.24·13-s + 9.76i·14-s + 0.838i·15-s − 3.61·16-s + 3.99i·17-s + ⋯ |
L(s) = 1 | + 1.44·2-s − 0.484i·3-s + 1.08·4-s − 0.447·5-s − 0.699i·6-s + 1.80i·7-s + 0.126·8-s + 0.765·9-s − 0.646·10-s + (−0.812 + 0.582i)11-s − 0.526i·12-s + 1.17·13-s + 2.61i·14-s + 0.216i·15-s − 0.904·16-s + 0.969i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.218463952\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.218463952\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 + (2.69 - 1.93i)T \) |
| 19 | \( 1 + (-4.34 + 0.391i)T \) |
good | 2 | \( 1 - 2.04T + 2T^{2} \) |
| 3 | \( 1 + 0.838iT - 3T^{2} \) |
| 7 | \( 1 - 4.78iT - 7T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 3.99iT - 17T^{2} \) |
| 23 | \( 1 - 5.80T + 23T^{2} \) |
| 29 | \( 1 + 0.280T + 29T^{2} \) |
| 31 | \( 1 - 1.69iT - 31T^{2} \) |
| 37 | \( 1 + 2.87iT - 37T^{2} \) |
| 41 | \( 1 + 5.74T + 41T^{2} \) |
| 43 | \( 1 + 5.09iT - 43T^{2} \) |
| 47 | \( 1 + 1.73T + 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 1.61iT - 59T^{2} \) |
| 61 | \( 1 - 1.50iT - 61T^{2} \) |
| 67 | \( 1 - 12.0iT - 67T^{2} \) |
| 71 | \( 1 - 12.3iT - 71T^{2} \) |
| 73 | \( 1 - 0.489iT - 73T^{2} \) |
| 79 | \( 1 - 6.76T + 79T^{2} \) |
| 83 | \( 1 + 14.7iT - 83T^{2} \) |
| 89 | \( 1 - 4.01iT - 89T^{2} \) |
| 97 | \( 1 + 11.2iT - 97T^{2} \) |
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
−10.14177356651590095359402173573, −8.983363619406231042364113068538, −8.344252209928020995456570319445, −7.22081776310145863015874884424, −6.40280331650079618784881822470, −5.52877287692375959595133760754, −4.96670149153386589345257018355, −3.79098429254575127953037224684, −2.87393670574332238962272978376, −1.80928058472577494107462699663,
0.988417674284797386855959856075, 3.20687150170921854038172402864, 3.59674060444604047361425867473, 4.58962379302767877268421347671, 5.05976406210046590141122951341, 6.33282237059308014030009572909, 7.17385998143192749299934410857, 7.81190244345196048888167642884, 9.117960414640203741315737277752, 10.07714897135536999999648716139