L(s) = 1 | + (2.28 − 3.95i)2-s + (−6.41 − 11.1i)4-s + (10.3 + 18.0i)5-s + (3.5 − 6.06i)7-s − 22.0·8-s + 94.8·10-s + (22.6 − 39.3i)11-s + (−14.6 − 25.3i)13-s + (−15.9 − 27.6i)14-s + (1.04 − 1.81i)16-s + 98.0·17-s + 31.1·19-s + (133. − 230. i)20-s + (−103. − 179. i)22-s + (4.19 + 7.26i)23-s + ⋯ |
L(s) = 1 | + (0.806 − 1.39i)2-s + (−0.801 − 1.38i)4-s + (0.929 + 1.61i)5-s + (0.188 − 0.327i)7-s − 0.973·8-s + 3.00·10-s + (0.622 − 1.07i)11-s + (−0.312 − 0.541i)13-s + (−0.304 − 0.528i)14-s + (0.0163 − 0.0283i)16-s + 1.39·17-s + 0.376·19-s + (1.49 − 2.58i)20-s + (−1.00 − 1.73i)22-s + (0.0380 + 0.0658i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0376 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0376 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.27145 - 2.18758i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.27145 - 2.18758i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-3.5 + 6.06i)T \) |
good | 2 | \( 1 + (-2.28 + 3.95i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-10.3 - 18.0i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-22.6 + 39.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (14.6 + 25.3i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 98.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-4.19 - 7.26i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (36.1 - 62.5i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-15.4 - 26.7i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-106. - 184. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (118. - 205. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (110. - 190. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 55.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + (327. + 566. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (174. - 302. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (105. + 182. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 548.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 266.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-134. + 233. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-312. + 541. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-145. + 252. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.67608273299375011566864352921, −10.96063107839544459993322784055, −10.27015919862094400827925379976, −9.532089086520855403475989408724, −7.64095142443614679199664995921, −6.28956904272525972407708421297, −5.30361256739707124590441537258, −3.49375895026817068323649150992, −2.92755198575027433565321772892, −1.40803236426816682296940331785,
1.62853322429164973360612516618, 4.18279758742032238357193203156, 5.11873000888560978627915622638, 5.75645136214512704186470172380, 7.00840082774838456010566787908, 8.136062604234353836457832335779, 9.126597625282542699107332778248, 9.926060750794621875328735967812, 12.14270865570122696783036113181, 12.44577645507283483112273449728