L(s) = 1 | + (−1.17 − 4.86i)5-s + (−5.87 − 3.39i)7-s + (8.57 + 4.94i)11-s + (17.6 − 10.1i)13-s + 19.1·17-s + 12·19-s + (−4.79 − 8.30i)23-s + (−22.2 + 11.4i)25-s + (−7.34 − 4.24i)29-s + (19 + 32.9i)31-s + (−9.59 + 32.5i)35-s + 6.78i·37-s + (−60.0 + 34.6i)41-s + (58.7 + 33.9i)43-s + (38.3 − 66.4i)47-s + ⋯ |
L(s) = 1 | + (−0.234 − 0.972i)5-s + (−0.839 − 0.484i)7-s + (0.779 + 0.449i)11-s + (1.35 − 0.782i)13-s + 1.12·17-s + 0.631·19-s + (−0.208 − 0.361i)23-s + (−0.889 + 0.456i)25-s + (−0.253 − 0.146i)29-s + (0.612 + 1.06i)31-s + (−0.274 + 0.929i)35-s + 0.183i·37-s + (−1.46 + 0.845i)41-s + (1.36 + 0.788i)43-s + (0.816 − 1.41i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0622 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0622 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.905804803\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.905804803\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.17 + 4.86i)T \) |
good | 7 | \( 1 + (5.87 + 3.39i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.57 - 4.94i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-17.6 + 10.1i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 19.1T + 289T^{2} \) |
| 19 | \( 1 - 12T + 361T^{2} \) |
| 23 | \( 1 + (4.79 + 8.30i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (7.34 + 4.24i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-19 - 32.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 6.78iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (60.0 - 34.6i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-58.7 - 33.9i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-38.3 + 66.4i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + (-72.2 + 41.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-35 + 60.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (93.9 - 54.2i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 13.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (15 - 25.9i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-67.1 + 116. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 32.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (82.2 + 47.4i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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
−8.956831814815197670335320782292, −8.310081822080323197378125755908, −7.49639142018870700570143652809, −6.54743086859605274583584083822, −5.77274237096315554420777936602, −4.86181380866191122732486501846, −3.79499118972699040799104465979, −3.26431311959624513699171379486, −1.45901529037273705395838906854, −0.64196217447151302228840326111,
1.10685167613841384126529542385, 2.53619467986412609031087062997, 3.53812925462567466032525516457, 3.97083095652081732862072371696, 5.70016223169246548194676023052, 6.10682942155193908008155256163, 6.92772510285084616339586298585, 7.71042078637330979391595827642, 8.739939932569556973640344900817, 9.368101320753904596876494057686