| L(s) = 1 | + (1.00 + 0.801i)2-s + (0.548 + 3.63i)3-s + (−0.522 − 2.28i)4-s + (0.680 + 0.997i)5-s + (−2.36 + 4.09i)6-s + (−2.49 + 1.44i)7-s + (3.54 − 7.35i)8-s + (−4.33 + 1.33i)9-s + (−0.116 + 1.54i)10-s + (2.00 − 8.79i)11-s + (8.03 − 3.15i)12-s + (−0.532 − 7.10i)13-s + (−3.66 − 0.553i)14-s + (−3.25 + 3.02i)15-s + (0.999 − 0.481i)16-s + (−10.5 − 7.18i)17-s + ⋯ |
| L(s) = 1 | + (0.502 + 0.400i)2-s + (0.182 + 1.21i)3-s + (−0.130 − 0.571i)4-s + (0.136 + 0.199i)5-s + (−0.394 + 0.682i)6-s + (−0.357 + 0.206i)7-s + (0.442 − 0.919i)8-s + (−0.481 + 0.148i)9-s + (−0.0116 + 0.154i)10-s + (0.182 − 0.799i)11-s + (0.669 − 0.262i)12-s + (−0.0409 − 0.546i)13-s + (−0.262 − 0.0395i)14-s + (−0.217 + 0.201i)15-s + (0.0624 − 0.0300i)16-s + (−0.620 − 0.422i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.833i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.553 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.21483 + 0.651620i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21483 + 0.651620i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-41.8 - 9.66i)T \) |
| good | 2 | \( 1 + (-1.00 - 0.801i)T + (0.890 + 3.89i)T^{2} \) |
| 3 | \( 1 + (-0.548 - 3.63i)T + (-8.60 + 2.65i)T^{2} \) |
| 5 | \( 1 + (-0.680 - 0.997i)T + (-9.13 + 23.2i)T^{2} \) |
| 7 | \( 1 + (2.49 - 1.44i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-2.00 + 8.79i)T + (-109. - 52.4i)T^{2} \) |
| 13 | \( 1 + (0.532 + 7.10i)T + (-167. + 25.1i)T^{2} \) |
| 17 | \( 1 + (10.5 + 7.18i)T + (105. + 269. i)T^{2} \) |
| 19 | \( 1 + (9.82 - 31.8i)T + (-298. - 203. i)T^{2} \) |
| 23 | \( 1 + (7.08 + 6.57i)T + (39.5 + 527. i)T^{2} \) |
| 29 | \( 1 + (5.07 - 33.6i)T + (-803. - 247. i)T^{2} \) |
| 31 | \( 1 + (-14.6 - 37.3i)T + (-704. + 653. i)T^{2} \) |
| 37 | \( 1 + (24.1 + 13.9i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-0.232 + 0.292i)T + (-374. - 1.63e3i)T^{2} \) |
| 47 | \( 1 + (14.6 + 64.0i)T + (-1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (-7.31 + 97.6i)T + (-2.77e3 - 418. i)T^{2} \) |
| 59 | \( 1 + (37.7 - 18.1i)T + (2.17e3 - 2.72e3i)T^{2} \) |
| 61 | \( 1 + (-32.1 - 12.5i)T + (2.72e3 + 2.53e3i)T^{2} \) |
| 67 | \( 1 + (-66.6 - 20.5i)T + (3.70e3 + 2.52e3i)T^{2} \) |
| 71 | \( 1 + (-26.4 - 28.4i)T + (-376. + 5.02e3i)T^{2} \) |
| 73 | \( 1 + (-22.3 + 1.67i)T + (5.26e3 - 794. i)T^{2} \) |
| 79 | \( 1 + (4.69 + 8.13i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-142. + 21.5i)T + (6.58e3 - 2.03e3i)T^{2} \) |
| 89 | \( 1 + (8.92 + 59.2i)T + (-7.56e3 + 2.33e3i)T^{2} \) |
| 97 | \( 1 + (25.3 - 111. i)T + (-8.47e3 - 4.08e3i)T^{2} \) |
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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
−15.90520808203894643729976762925, −14.73655877795721407888547310764, −14.09117145244133043684267248183, −12.62403102237200493306002864420, −10.69765235794249961661101738569, −10.01388145089356144089224232666, −8.712303177512739522324243749128, −6.48445786381908028778254687229, −5.17505562073857996682062719905, −3.68875022875112661262542344608,
2.26757179210218087341650772701, 4.40546673915176591699095301487, 6.67020498871332939214961037751, 7.78045748469532619206420834896, 9.238169266080401911132423051917, 11.23693192280845388219016494866, 12.37606673359763896396738835873, 13.16615104570854194455078739885, 13.77597700451059039692652523257, 15.38368798665498752122008150464
