| L(s) = 1 | + (−1.12 − 2.34i)2-s + (12.1 − 25.2i)3-s + (35.6 − 44.7i)4-s + (−135. − 30.9i)5-s − 73.0·6-s − 261. i·7-s + (−307. − 70.2i)8-s + (−35.4 − 44.5i)9-s + (80.5 + 352. i)10-s + (582. + 730. i)11-s + (−695. − 1.44e3i)12-s + (−361. + 1.58e3i)13-s + (−614. + 295. i)14-s + (−2.42e3 + 3.04e3i)15-s + (−631. − 2.76e3i)16-s + (−322. − 1.41e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.141 − 0.293i)2-s + (0.450 − 0.935i)3-s + (0.557 − 0.698i)4-s + (−1.08 − 0.247i)5-s − 0.338·6-s − 0.763i·7-s + (−0.601 − 0.137i)8-s + (−0.0486 − 0.0610i)9-s + (0.0805 + 0.352i)10-s + (0.437 + 0.548i)11-s + (−0.402 − 0.836i)12-s + (−0.164 + 0.721i)13-s + (−0.223 + 0.107i)14-s + (−0.719 + 0.902i)15-s + (−0.154 − 0.675i)16-s + (−0.0656 − 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0741749 - 1.38841i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0741749 - 1.38841i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.40e4 - 4.71e4i)T \) |
| good | 2 | \( 1 + (1.12 + 2.34i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (-12.1 + 25.2i)T + (-454. - 569. i)T^{2} \) |
| 5 | \( 1 + (135. + 30.9i)T + (1.40e4 + 6.77e3i)T^{2} \) |
| 7 | \( 1 + 261. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + (-582. - 730. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (361. - 1.58e3i)T + (-4.34e6 - 2.09e6i)T^{2} \) |
| 17 | \( 1 + (322. + 1.41e3i)T + (-2.17e7 + 1.04e7i)T^{2} \) |
| 19 | \( 1 + (1.01e4 + 8.11e3i)T + (1.04e7 + 4.58e7i)T^{2} \) |
| 23 | \( 1 + (-4.29e3 - 5.38e3i)T + (-3.29e7 + 1.44e8i)T^{2} \) |
| 29 | \( 1 + (-1.70e3 - 3.53e3i)T + (-3.70e8 + 4.65e8i)T^{2} \) |
| 31 | \( 1 + (-1.66e4 + 8.02e3i)T + (5.53e8 - 6.93e8i)T^{2} \) |
| 37 | \( 1 + 8.99e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-1.12e5 + 5.41e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-3.04e4 + 3.82e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (8.58e3 + 3.75e4i)T + (-1.99e10 + 9.61e9i)T^{2} \) |
| 59 | \( 1 + (-2.09e4 - 9.18e4i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-8.00e4 + 1.66e5i)T + (-3.21e10 - 4.02e10i)T^{2} \) |
| 67 | \( 1 + (1.28e5 - 1.60e5i)T + (-2.01e10 - 8.81e10i)T^{2} \) |
| 71 | \( 1 + (2.86e5 + 2.28e5i)T + (2.85e10 + 1.24e11i)T^{2} \) |
| 73 | \( 1 + (-7.92e4 - 1.80e4i)T + (1.36e11 + 6.56e10i)T^{2} \) |
| 79 | \( 1 - 1.99e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (-1.37e5 - 6.64e4i)T + (2.03e11 + 2.55e11i)T^{2} \) |
| 89 | \( 1 + (-2.19e4 + 4.54e4i)T + (-3.09e11 - 3.88e11i)T^{2} \) |
| 97 | \( 1 + (-8.98e5 - 1.12e6i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.12893058416186272904634704919, −12.83938585652623478484841662234, −11.73770377641158499934689226057, −10.70050369802751863096155160633, −9.073962035485037870327223860024, −7.50185355100965209809728390954, −6.75088420275231438776920202568, −4.34088285096015569222205700278, −2.17172174208639037627348453467, −0.64140730897992308735272487621,
2.98653950716684876175944455513, 4.09364486084191249374847617017, 6.33645664684734240853431022915, 8.011898440858292619784512686966, 8.759129418943123793915471932475, 10.45408144126308349986840130957, 11.71023326354889809909240103908, 12.63178004705441409230767022591, 14.79733306189129257704120409221, 15.26441472489473735847505344566
