L(s) = 1 | + (−0.908 + 0.417i)2-s + (0.739 − 0.673i)3-s + (0.650 − 0.759i)4-s + (−0.998 − 0.0615i)5-s + (−0.389 + 0.920i)6-s + (0.969 + 0.243i)7-s + (−0.273 + 0.961i)8-s + (0.0922 − 0.995i)9-s + (0.932 − 0.361i)10-s + (0.816 + 0.577i)11-s + (−0.0307 − 0.999i)12-s + (−0.273 − 0.961i)13-s + (−0.982 + 0.183i)14-s + (−0.779 + 0.626i)15-s + (−0.153 − 0.988i)16-s + (−0.389 − 0.920i)17-s + ⋯ |
L(s) = 1 | + (−0.908 + 0.417i)2-s + (0.739 − 0.673i)3-s + (0.650 − 0.759i)4-s + (−0.998 − 0.0615i)5-s + (−0.389 + 0.920i)6-s + (0.969 + 0.243i)7-s + (−0.273 + 0.961i)8-s + (0.0922 − 0.995i)9-s + (0.932 − 0.361i)10-s + (0.816 + 0.577i)11-s + (−0.0307 − 0.999i)12-s + (−0.273 − 0.961i)13-s + (−0.982 + 0.183i)14-s + (−0.779 + 0.626i)15-s + (−0.153 − 0.988i)16-s + (−0.389 − 0.920i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7815911821 - 0.2440931392i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7815911821 - 0.2440931392i\) |
\(L(1)\) |
\(\approx\) |
\(0.8408152132 - 0.1192511482i\) |
\(L(1)\) |
\(\approx\) |
\(0.8408152132 - 0.1192511482i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 103 | \( 1 \) |
good | 2 | \( 1 + (-0.908 + 0.417i)T \) |
| 3 | \( 1 + (0.739 - 0.673i)T \) |
| 5 | \( 1 + (-0.998 - 0.0615i)T \) |
| 7 | \( 1 + (0.969 + 0.243i)T \) |
| 11 | \( 1 + (0.816 + 0.577i)T \) |
| 13 | \( 1 + (-0.273 - 0.961i)T \) |
| 17 | \( 1 + (-0.389 - 0.920i)T \) |
| 19 | \( 1 + (0.213 - 0.976i)T \) |
| 23 | \( 1 + (0.0922 + 0.995i)T \) |
| 29 | \( 1 + (0.552 - 0.833i)T \) |
| 31 | \( 1 + (0.932 + 0.361i)T \) |
| 37 | \( 1 + (-0.850 + 0.526i)T \) |
| 41 | \( 1 + (-0.998 + 0.0615i)T \) |
| 43 | \( 1 + (-0.0307 + 0.999i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.952 + 0.303i)T \) |
| 59 | \( 1 + (0.969 - 0.243i)T \) |
| 61 | \( 1 + (-0.602 + 0.798i)T \) |
| 67 | \( 1 + (-0.696 - 0.717i)T \) |
| 71 | \( 1 + (0.552 + 0.833i)T \) |
| 73 | \( 1 + (0.445 - 0.895i)T \) |
| 79 | \( 1 + (0.445 + 0.895i)T \) |
| 83 | \( 1 + (-0.696 + 0.717i)T \) |
| 89 | \( 1 + (-0.982 + 0.183i)T \) |
| 97 | \( 1 + (-0.389 + 0.920i)T \) |
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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
−30.10153168166861974939808034952, −28.53247138571512006992254821399, −27.5332075975496488474708084527, −26.91534152532274487002406939961, −26.329845238026507275802141062511, −24.88209164130662836353568376023, −24.0412574117577938308339316513, −22.240400912109100723974577770368, −21.23676684437785151360988233666, −20.33995037691684296028929378991, −19.443765824410767842167066754563, −18.70373612384745062645158260087, −17.05012652303282790971202175647, −16.278621243414442812783900495350, −15.0549722265279802711293072743, −14.096312974113144925806059260077, −12.176539428713502062912777312553, −11.203721401502253288218139091923, −10.268719719430096503746303016378, −8.71029586553536176448776056093, −8.273967836317709313034663929836, −6.9395350477778220275800533596, −4.39996324287882797976379611382, −3.49888257954276652427094747426, −1.77493264525356585381585466402,
1.202940320529788322772915683208, 2.81486259102884928535986806367, 4.83545350742589358484781185716, 6.77252770582006492660730940903, 7.68057311164400714301010364854, 8.46317125266511429813620268180, 9.57406828618200101748290919439, 11.333545227306773537815530419431, 12.0988047169347121672308887340, 13.88756565994744432119988408729, 15.08877264105796151546962444974, 15.55565138043539661203246700413, 17.4094330649015888850935724140, 18.00980373793584046839961173979, 19.28972475714131529867955145360, 19.924710455553876662096563020776, 20.77979760903962241975983258597, 22.815735025452654427944248629192, 23.97269327548947875791927311625, 24.669048132233424942584456801611, 25.44813092443886675586373934165, 26.81127080235130840244396715595, 27.38161638483488398673747366336, 28.38163976677337859918944747646, 29.876686432187697768563808449500