L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.0747 − 0.997i)3-s + (0.623 + 0.781i)4-s + (0.955 + 0.294i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + (−0.222 − 0.974i)8-s + (−0.988 − 0.149i)9-s + (−0.733 − 0.680i)10-s + (0.623 − 0.781i)11-s + (0.826 − 0.563i)12-s + (−0.733 + 0.680i)13-s + (0.0747 + 0.997i)14-s + (0.365 − 0.930i)15-s + (−0.222 + 0.974i)16-s + (0.955 − 0.294i)17-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.0747 − 0.997i)3-s + (0.623 + 0.781i)4-s + (0.955 + 0.294i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + (−0.222 − 0.974i)8-s + (−0.988 − 0.149i)9-s + (−0.733 − 0.680i)10-s + (0.623 − 0.781i)11-s + (0.826 − 0.563i)12-s + (−0.733 + 0.680i)13-s + (0.0747 + 0.997i)14-s + (0.365 − 0.930i)15-s + (−0.222 + 0.974i)16-s + (0.955 − 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4647456281 - 0.4125800977i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4647456281 - 0.4125800977i\) |
\(L(1)\) |
\(\approx\) |
\(0.6603551965 - 0.3590084349i\) |
\(L(1)\) |
\(\approx\) |
\(0.6603551965 - 0.3590084349i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (-0.900 - 0.433i)T \) |
| 3 | \( 1 + (0.0747 - 0.997i)T \) |
| 5 | \( 1 + (0.955 + 0.294i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.623 - 0.781i)T \) |
| 13 | \( 1 + (-0.733 + 0.680i)T \) |
| 17 | \( 1 + (0.955 - 0.294i)T \) |
| 19 | \( 1 + (-0.988 + 0.149i)T \) |
| 23 | \( 1 + (0.365 + 0.930i)T \) |
| 29 | \( 1 + (0.0747 + 0.997i)T \) |
| 31 | \( 1 + (0.826 - 0.563i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.900 - 0.433i)T \) |
| 47 | \( 1 + (0.623 + 0.781i)T \) |
| 53 | \( 1 + (-0.733 - 0.680i)T \) |
| 59 | \( 1 + (-0.222 + 0.974i)T \) |
| 61 | \( 1 + (0.826 + 0.563i)T \) |
| 67 | \( 1 + (-0.988 + 0.149i)T \) |
| 71 | \( 1 + (0.365 - 0.930i)T \) |
| 73 | \( 1 + (-0.733 + 0.680i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.0747 - 0.997i)T \) |
| 89 | \( 1 + (0.0747 - 0.997i)T \) |
| 97 | \( 1 + (0.623 - 0.781i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−34.70651914803993834814203265428, −33.84117810965259355426538692474, −32.62162335276769186292316343966, −32.086449736243884680616169741882, −29.88641126573817020740756206656, −28.454671427468243347198427784097, −27.998560976698507234100901033457, −26.62399560550103286622030143233, −25.375468235018114233102616286457, −24.98878818738886948443858611854, −22.87630140972969784904672893905, −21.586222299044942503364176391436, −20.43273101412197387879575396967, −19.15445271150432604234836017743, −17.51906626970687980730802815503, −16.77748760978067799356689358592, −15.37321869537416553516807337995, −14.49050971330321669148209288696, −12.286612794887876946930223536935, −10.33426627788428484192681114514, −9.59050271152605424222760583917, −8.51486375665073543892518496365, −6.3296563583367392518086166111, −5.10947782001000455003081293174, −2.444158638839448889293888696202,
1.44643056412635985582160308848, 3.107946651971263010221201775127, 6.3213604143417173938116238408, 7.31305247199816800727266678047, 8.97501963009031004521120379724, 10.26104378733710409258580834583, 11.725844076293417977040819977026, 13.130440657276665531200145919468, 14.24664543823765243587339508451, 16.775941041297409202257142239441, 17.30197422967764616939031579924, 18.80726133898896658320065346965, 19.45530800265856741268402216955, 20.89917085603887733904883024926, 22.238483152538161247394752145514, 23.92748036163812102851333689699, 25.257659007015509623813077316381, 25.96384633585835919981122582488, 27.25387571323338205977018841748, 28.96654594099520994699020470348, 29.59091876293576439950561775522, 30.17800701376258662949445326536, 31.877335689046355581458917017423, 33.58227919483154602615751645325, 34.61480111203549865752260874663