L(s) = 1 | + (0.739 − 0.673i)2-s + (−0.932 + 0.361i)3-s + (0.0922 − 0.995i)4-s + (0.850 + 0.526i)5-s + (−0.445 + 0.895i)6-s + (−0.602 + 0.798i)7-s + (−0.602 − 0.798i)8-s + (0.739 − 0.673i)9-s + (0.982 − 0.183i)10-s + (−0.739 + 0.673i)11-s + (0.273 + 0.961i)12-s + (−0.602 + 0.798i)13-s + (0.0922 + 0.995i)14-s + (−0.982 − 0.183i)15-s + (−0.982 − 0.183i)16-s + (0.445 + 0.895i)17-s + ⋯ |
L(s) = 1 | + (0.739 − 0.673i)2-s + (−0.932 + 0.361i)3-s + (0.0922 − 0.995i)4-s + (0.850 + 0.526i)5-s + (−0.445 + 0.895i)6-s + (−0.602 + 0.798i)7-s + (−0.602 − 0.798i)8-s + (0.739 − 0.673i)9-s + (0.982 − 0.183i)10-s + (−0.739 + 0.673i)11-s + (0.273 + 0.961i)12-s + (−0.602 + 0.798i)13-s + (0.0922 + 0.995i)14-s + (−0.982 − 0.183i)15-s + (−0.982 − 0.183i)16-s + (0.445 + 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.333952926 + 0.7740859645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.333952926 + 0.7740859645i\) |
\(L(1)\) |
\(\approx\) |
\(1.167427487 + 0.04754516645i\) |
\(L(1)\) |
\(\approx\) |
\(1.167427487 + 0.04754516645i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 103 | \( 1 \) |
good | 2 | \( 1 + (0.739 - 0.673i)T \) |
| 3 | \( 1 + (-0.932 + 0.361i)T \) |
| 5 | \( 1 + (0.850 + 0.526i)T \) |
| 7 | \( 1 + (-0.602 + 0.798i)T \) |
| 11 | \( 1 + (-0.739 + 0.673i)T \) |
| 13 | \( 1 + (-0.602 + 0.798i)T \) |
| 17 | \( 1 + (0.445 + 0.895i)T \) |
| 19 | \( 1 + (0.932 + 0.361i)T \) |
| 23 | \( 1 + (0.739 + 0.673i)T \) |
| 29 | \( 1 + (-0.850 - 0.526i)T \) |
| 31 | \( 1 + (0.982 + 0.183i)T \) |
| 37 | \( 1 + (0.273 + 0.961i)T \) |
| 41 | \( 1 + (-0.850 + 0.526i)T \) |
| 43 | \( 1 + (0.273 - 0.961i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.932 - 0.361i)T \) |
| 59 | \( 1 + (-0.602 - 0.798i)T \) |
| 61 | \( 1 + (0.445 + 0.895i)T \) |
| 67 | \( 1 + (0.602 + 0.798i)T \) |
| 71 | \( 1 + (0.850 - 0.526i)T \) |
| 73 | \( 1 + (0.850 - 0.526i)T \) |
| 79 | \( 1 + (-0.850 - 0.526i)T \) |
| 83 | \( 1 + (-0.602 + 0.798i)T \) |
| 89 | \( 1 + (-0.0922 - 0.995i)T \) |
| 97 | \( 1 + (0.445 - 0.895i)T \) |
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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
−29.475796802321431432216576027821, −28.77132629365928981295022834742, −27.1633244483531163814321482127, −26.10652423523439777245750828342, −24.818732339525939108070614652063, −24.28875505781772937471105654871, −23.08204932107411119610888481055, −22.422306449195904938141649525481, −21.33217126848717162501192816597, −20.2636823246455287913104450003, −18.43129079314554484798943352909, −17.38759318956062460632172743594, −16.602630046494998476505038274311, −15.852972618847228202059531189682, −14.03116335070577546018731838582, −13.18127031209366629028276715401, −12.50582619337719241333483908818, −11.01124564542638396164811130462, −9.67821855741773710741647254863, −7.835641253760088651611908062, −6.77776235474451823225266439229, −5.585594928032195659520347936843, −4.86718930511585895075621830556, −2.914529416978122938120193426621, −0.584003817444251988330973679340,
1.78880264929438783804243831104, 3.23207336391542348007200615939, 4.94274765222361169106789175176, 5.810755067326185486822497906224, 6.83321517876833774442242690797, 9.63567310369743252331019339657, 10.01190059645165147305199152292, 11.37107725725056771725916010147, 12.35390999190894127108124254283, 13.321536754846995404483375990361, 14.78701571927289735839280729468, 15.6129552265859182724524818618, 17.0676923077626696780032919707, 18.32884315949137966594921793480, 19.06989989116613589510151537855, 20.81416571259632502590994055578, 21.58307549347925642846924914030, 22.301066282962855133569123481615, 23.11102667873781865176822252373, 24.25772656410292820393639460336, 25.52806416836073407429251184868, 26.79497082158661305895104512698, 28.27422955952886919793245495356, 28.77234363469079440918198218789, 29.47446376160235140468095187191