L(s) = 1 | + (0.273 − 0.961i)2-s + (0.183 − 0.982i)3-s + (−0.850 − 0.526i)4-s + (−0.961 + 0.273i)5-s + (−0.895 − 0.445i)6-s + (−0.0922 − 0.995i)7-s + (−0.739 + 0.673i)8-s + (−0.932 − 0.361i)9-s + i·10-s + (0.850 + 0.526i)11-s + (−0.673 + 0.739i)12-s + (−0.995 + 0.0922i)13-s + (−0.982 − 0.183i)14-s + (0.0922 + 0.995i)15-s + (0.445 + 0.895i)16-s + (−0.739 − 0.673i)17-s + ⋯ |
L(s) = 1 | + (0.273 − 0.961i)2-s + (0.183 − 0.982i)3-s + (−0.850 − 0.526i)4-s + (−0.961 + 0.273i)5-s + (−0.895 − 0.445i)6-s + (−0.0922 − 0.995i)7-s + (−0.739 + 0.673i)8-s + (−0.932 − 0.361i)9-s + i·10-s + (0.850 + 0.526i)11-s + (−0.673 + 0.739i)12-s + (−0.995 + 0.0922i)13-s + (−0.982 − 0.183i)14-s + (0.0922 + 0.995i)15-s + (0.445 + 0.895i)16-s + (−0.739 − 0.673i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1692313395 - 0.7307380922i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1692313395 - 0.7307380922i\) |
\(L(1)\) |
\(\approx\) |
\(0.4529300230 - 0.7277431858i\) |
\(L(1)\) |
\(\approx\) |
\(0.4529300230 - 0.7277431858i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 137 | \( 1 \) |
good | 2 | \( 1 + (0.273 - 0.961i)T \) |
| 3 | \( 1 + (0.183 - 0.982i)T \) |
| 5 | \( 1 + (-0.961 + 0.273i)T \) |
| 7 | \( 1 + (-0.0922 - 0.995i)T \) |
| 11 | \( 1 + (0.850 + 0.526i)T \) |
| 13 | \( 1 + (-0.995 + 0.0922i)T \) |
| 17 | \( 1 + (-0.739 - 0.673i)T \) |
| 19 | \( 1 + (0.602 - 0.798i)T \) |
| 23 | \( 1 + (-0.895 + 0.445i)T \) |
| 29 | \( 1 + (0.895 - 0.445i)T \) |
| 31 | \( 1 + (0.798 - 0.602i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (-0.798 - 0.602i)T \) |
| 47 | \( 1 + (0.361 - 0.932i)T \) |
| 53 | \( 1 + (0.798 + 0.602i)T \) |
| 59 | \( 1 + (0.932 + 0.361i)T \) |
| 61 | \( 1 + (-0.932 + 0.361i)T \) |
| 67 | \( 1 + (0.995 - 0.0922i)T \) |
| 71 | \( 1 + (-0.526 - 0.850i)T \) |
| 73 | \( 1 + (0.0922 - 0.995i)T \) |
| 79 | \( 1 + (-0.183 - 0.982i)T \) |
| 83 | \( 1 + (0.673 + 0.739i)T \) |
| 89 | \( 1 + (0.961 - 0.273i)T \) |
| 97 | \( 1 + (0.526 - 0.850i)T \) |
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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
−28.65612979914801940870058186650, −27.64433833385412002715463184756, −27.033801884139014462886594644443, −26.21964342583531895088812391049, −24.89209526222757416343650789101, −24.37743167425519835884094875375, −22.957681682378616851700255705136, −22.15321971535304235830565705107, −21.49540147284140012520104500294, −19.98792771475517407961835216658, −19.06635314059016688768258142825, −17.58033613967286944564325326731, −16.44303264699588233425540954072, −15.81756475765588136510926390423, −14.91871975644595922702101283788, −14.17864811663744010572722189666, −12.44876216902440230429350379051, −11.66228504510659380233097760961, −9.90169561307910404433228830252, −8.72588365684851876972907984890, −8.1600769532600335217937033756, −6.47286892059297183413390081309, −5.21082405872141706540075609928, −4.22174505657767835461159804401, −3.11852187625920370924806484986,
0.60648496489733105278161590854, 2.27030889183531427670450779535, 3.59340116234635353419375094659, 4.72044159954461745406916685307, 6.711148477014540012358441186487, 7.56994755931491795736793732668, 8.9866805061076397013290425131, 10.295633221844805359906765654317, 11.74322099958396545067058018974, 12.00178081610014103591510109615, 13.47549054589497843791914406271, 14.136685068752079489969818799188, 15.315264026136443352088911607140, 17.1670956215324843460815495686, 17.99791818646696435465196524936, 19.29792009125864188763901938515, 19.79775378477091606631395933908, 20.39537409909623008211593371589, 22.2386715287000280596692211990, 22.84763808100783846771925602209, 23.81128432678199264955980435196, 24.551245847625496343659499905883, 26.26176308645225541669990328390, 26.99160024923613675495133439393, 28.09127479025539181400271243892