| L(s) = 1 | + (−0.716 − 1.48i)2-s + (3.40 − 7.06i)3-s + (38.2 − 47.9i)4-s + (69.5 + 15.8i)5-s − 12.9·6-s − 168. i·7-s + (−201. − 46.0i)8-s + (416. + 521. i)9-s + (−26.2 − 114. i)10-s + (−494. − 619. i)11-s + (−208. − 432. i)12-s + (419. − 1.83e3i)13-s + (−250. + 120. i)14-s + (348. − 437. i)15-s + (−796. − 3.49e3i)16-s + (−1.48e3 − 6.51e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.0895 − 0.185i)2-s + (0.125 − 0.261i)3-s + (0.596 − 0.748i)4-s + (0.556 + 0.126i)5-s − 0.0598·6-s − 0.490i·7-s + (−0.393 − 0.0898i)8-s + (0.570 + 0.715i)9-s + (−0.0262 − 0.114i)10-s + (−0.371 − 0.465i)11-s + (−0.120 − 0.250i)12-s + (0.190 − 0.836i)13-s + (−0.0911 + 0.0438i)14-s + (0.103 − 0.129i)15-s + (−0.194 − 0.852i)16-s + (−0.302 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0601 + 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0601 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.50274 - 1.41493i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50274 - 1.41493i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-4.41e4 - 6.61e4i)T \) |
| good | 2 | \( 1 + (0.716 + 1.48i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (-3.40 + 7.06i)T + (-454. - 569. i)T^{2} \) |
| 5 | \( 1 + (-69.5 - 15.8i)T + (1.40e4 + 6.77e3i)T^{2} \) |
| 7 | \( 1 + 168. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + (494. + 619. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-419. + 1.83e3i)T + (-4.34e6 - 2.09e6i)T^{2} \) |
| 17 | \( 1 + (1.48e3 + 6.51e3i)T + (-2.17e7 + 1.04e7i)T^{2} \) |
| 19 | \( 1 + (-7.48e3 - 5.97e3i)T + (1.04e7 + 4.58e7i)T^{2} \) |
| 23 | \( 1 + (1.07e4 + 1.35e4i)T + (-3.29e7 + 1.44e8i)T^{2} \) |
| 29 | \( 1 + (-1.61e4 - 3.35e4i)T + (-3.70e8 + 4.65e8i)T^{2} \) |
| 31 | \( 1 + (-2.24e4 + 1.07e4i)T + (5.53e8 - 6.93e8i)T^{2} \) |
| 37 | \( 1 - 9.35e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (6.82e4 - 3.28e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-4.19e4 + 5.25e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (295. + 1.29e3i)T + (-1.99e10 + 9.61e9i)T^{2} \) |
| 59 | \( 1 + (-6.05e4 - 2.65e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.15e5 + 2.39e5i)T + (-3.21e10 - 4.02e10i)T^{2} \) |
| 67 | \( 1 + (-1.13e5 + 1.42e5i)T + (-2.01e10 - 8.81e10i)T^{2} \) |
| 71 | \( 1 + (-1.20e5 - 9.59e4i)T + (2.85e10 + 1.24e11i)T^{2} \) |
| 73 | \( 1 + (-1.27e5 - 2.90e4i)T + (1.36e11 + 6.56e10i)T^{2} \) |
| 79 | \( 1 + 5.74e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (-5.36e5 - 2.58e5i)T + (2.03e11 + 2.55e11i)T^{2} \) |
| 89 | \( 1 + (2.00e5 - 4.17e5i)T + (-3.09e11 - 3.88e11i)T^{2} \) |
| 97 | \( 1 + (-8.47e5 - 1.06e6i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.16033955683432791778004620432, −13.50747724311876356304238148841, −11.92515320026510535621596902912, −10.51786193487771576874191559000, −9.937057198938225373843485188625, −7.959176072093545991617712670224, −6.59817393211411798845927680157, −5.18447459847552498535853817252, −2.67318972354813886979677230058, −1.05649366757799954941392168049,
2.07156134715766223259183320398, 3.90748489639895232601282635485, 5.97996690633988304406159655637, 7.30624726356636699729511055300, 8.813418289293240926462729150679, 9.925541139609405845418888907567, 11.57743814067416185772631752986, 12.52855156724890158334720712742, 13.76111448105357567663607889616, 15.44524398221767959751113358000
