| L(s) = 1 | + (−5.07 + 1.36i)2-s + (4.03 + 3.27i)3-s + (17.0 − 9.82i)4-s + (0.833 + 0.833i)5-s + (−24.9 − 11.1i)6-s + (−6.38 + 23.8i)7-s + (−43.3 + 43.3i)8-s + (5.58 + 26.4i)9-s + (−5.36 − 3.09i)10-s + (−2.47 − 9.25i)11-s + (100. + 16.0i)12-s + (−46.5 + 5.46i)13-s − 129. i·14-s + (0.636 + 6.08i)15-s + (82.4 − 142. i)16-s + (52.2 + 90.4i)17-s + ⋯ |
| L(s) = 1 | + (−1.79 + 0.481i)2-s + (0.776 + 0.629i)3-s + (2.12 − 1.22i)4-s + (0.0745 + 0.0745i)5-s + (−1.69 − 0.757i)6-s + (−0.344 + 1.28i)7-s + (−1.91 + 1.91i)8-s + (0.206 + 0.978i)9-s + (−0.169 − 0.0979i)10-s + (−0.0679 − 0.253i)11-s + (2.42 + 0.385i)12-s + (−0.993 + 0.116i)13-s − 2.47i·14-s + (0.0109 + 0.104i)15-s + (1.28 − 2.23i)16-s + (0.745 + 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.363969 + 0.576351i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.363969 + 0.576351i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.03 - 3.27i)T \) |
| 13 | \( 1 + (46.5 - 5.46i)T \) |
| good | 2 | \( 1 + (5.07 - 1.36i)T + (6.92 - 4i)T^{2} \) |
| 5 | \( 1 + (-0.833 - 0.833i)T + 125iT^{2} \) |
| 7 | \( 1 + (6.38 - 23.8i)T + (-297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (2.47 + 9.25i)T + (-1.15e3 + 665.5i)T^{2} \) |
| 17 | \( 1 + (-52.2 - 90.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-75.4 - 20.2i)T + (5.94e3 + 3.42e3i)T^{2} \) |
| 23 | \( 1 + (-36.9 + 63.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (44.3 + 25.6i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-61.5 + 61.5i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (-22.8 + 6.11i)T + (4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-241. + 64.8i)T + (5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-275. + 158. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-210. + 210. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 - 207. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (443. + 118. i)T + (1.77e5 + 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-179. - 310. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (12.3 + 45.9i)T + (-2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (-171. + 641. i)T + (-3.09e5 - 1.78e5i)T^{2} \) |
| 73 | \( 1 + (-9.85 - 9.85i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 740.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (501. + 501. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (-270. - 1.00e3i)T + (-6.10e5 + 3.52e5i)T^{2} \) |
| 97 | \( 1 + (-1.17e3 - 315. i)T + (7.90e5 + 4.56e5i)T^{2} \) |
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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
−16.18023904116681214370971472197, −15.32261734039579687476109691443, −14.48075512697642044856689113870, −12.19152961821969155914875522343, −10.54524546628669204065677408095, −9.606147051931925740380733332020, −8.723339860463403372129120160649, −7.67185724943792269622284439814, −5.84850849457460857445406725227, −2.45966710036515208691340455291,
0.969106540183531554056992714704, 2.99747882155030606573725149873, 7.26110542408330406558873441104, 7.52371323400626901484387447273, 9.306217445593498357766680540894, 9.927733810242579499191369231771, 11.45198903436254978486480261657, 12.73007070205926827704623798957, 14.13213311126819673365586616121, 15.80394049315024467976654505536
