| L(s) = 1 | + (1.67 − 1.08i)2-s + (−2.08 + 2.15i)3-s + (1.62 − 3.65i)4-s + (5.42 − 3.12i)5-s + (−1.15 + 5.88i)6-s + (5.96 + 3.44i)7-s + (−1.24 − 7.90i)8-s + (−0.284 − 8.99i)9-s + (5.68 − 11.1i)10-s + (−5.38 + 9.32i)11-s + (4.47 + 11.1i)12-s + (−15.3 + 8.88i)13-s + (13.7 − 0.719i)14-s + (−4.57 + 18.2i)15-s + (−10.7 − 11.8i)16-s − 0.681·17-s + ⋯ |
| L(s) = 1 | + (0.838 − 0.544i)2-s + (−0.695 + 0.718i)3-s + (0.406 − 0.913i)4-s + (1.08 − 0.625i)5-s + (−0.192 + 0.981i)6-s + (0.851 + 0.491i)7-s + (−0.156 − 0.987i)8-s + (−0.0316 − 0.999i)9-s + (0.568 − 1.11i)10-s + (−0.489 + 0.847i)11-s + (0.372 + 0.927i)12-s + (−1.18 + 0.683i)13-s + (0.982 − 0.0513i)14-s + (−0.304 + 1.21i)15-s + (−0.668 − 0.743i)16-s − 0.0400·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.72620 - 0.449664i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72620 - 0.449664i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.67 + 1.08i)T \) |
| 3 | \( 1 + (2.08 - 2.15i)T \) |
| good | 5 | \( 1 + (-5.42 + 3.12i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-5.96 - 3.44i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (5.38 - 9.32i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (15.3 - 8.88i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 0.681T + 289T^{2} \) |
| 19 | \( 1 + 26.6T + 361T^{2} \) |
| 23 | \( 1 + (-22.8 + 13.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-26.8 - 15.5i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (19.9 - 11.4i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 33.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-13.4 - 23.3i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (16.5 - 28.6i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (10.6 + 6.13i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 49.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (19.0 + 32.9i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-65.7 - 37.9i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (27.4 + 47.5i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 35.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 113.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-30.8 - 17.7i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-58.7 + 101. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 58.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + (63.7 - 110. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.51432821330500410756972261489, −12.90418762962380181753176895558, −12.25475954680514255000103381814, −11.02879799184065089934873926391, −10.03086099315353594097322825968, −9.079169531865818610325037783806, −6.60069460329751243480238124250, −5.15622045172415434245915456063, −4.69749388445573683029185081439, −2.09056077561751818441828156920,
2.44706572827351132454205567381, 4.95493068566612388631894466140, 5.96864101502894160766647267251, 7.08834644015410653186564328451, 8.177203400320177570801474405323, 10.44237477823275674629527955941, 11.25651859755317622880387930028, 12.60844293516448833662612290027, 13.50718348269032205096846266405, 14.23245477419141382658098366098
