L(s) = 1 | + 1.80·2-s − 4.72·4-s + (−1.04 + 1.81i)5-s + (18.3 + 2.47i)7-s − 23.0·8-s + (−1.89 + 3.28i)10-s + (11.7 + 20.3i)11-s + (27.8 + 48.2i)13-s + (33.2 + 4.47i)14-s − 3.83·16-s + (−55.3 + 95.9i)17-s + (9.75 + 16.8i)19-s + (4.95 − 8.58i)20-s + (21.2 + 36.7i)22-s + (−6.87 + 11.9i)23-s + ⋯ |
L(s) = 1 | + 0.639·2-s − 0.590·4-s + (−0.0938 + 0.162i)5-s + (0.991 + 0.133i)7-s − 1.01·8-s + (−0.0600 + 0.103i)10-s + (0.321 + 0.557i)11-s + (0.594 + 1.02i)13-s + (0.633 + 0.0854i)14-s − 0.0599·16-s + (−0.790 + 1.36i)17-s + (0.117 + 0.203i)19-s + (0.0554 − 0.0960i)20-s + (0.205 + 0.356i)22-s + (−0.0623 + 0.107i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.47209 + 1.15242i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47209 + 1.15242i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-18.3 - 2.47i)T \) |
good | 2 | \( 1 - 1.80T + 8T^{2} \) |
| 5 | \( 1 + (1.04 - 1.81i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-11.7 - 20.3i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-27.8 - 48.2i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (55.3 - 95.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-9.75 - 16.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (6.87 - 11.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (59.9 - 103. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 158.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (79.4 + 137. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (208. + 361. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-131. + 227. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 190.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (175. - 304. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 - 127.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 724.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 998.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 404.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (120. - 208. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + 921.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-502. + 869. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-239. - 414. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-431. + 747. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.37799573038711145600224038525, −11.52198506149737006412725762483, −10.51182911930871826540576761445, −9.086081077280104986883288424713, −8.486936451674154782417600273625, −7.02309155816194220981336326404, −5.77905479136525683688763779950, −4.60114870790275812461667088461, −3.75533225653161792477960069290, −1.75274961016373985176857385239,
0.73144301628962538661266506922, 2.97908725320663553029458366857, 4.40054759048706036428965974317, 5.18933928791661629701077973623, 6.42498560213409856586896270040, 8.039931495812313924870450415810, 8.705405403770534163249145113246, 9.905708503088853966552297940463, 11.21865208655221579701426792403, 11.88833115028068099009390111527