L(s) = 1 | + (0.881 − 0.473i)2-s + (−0.273 + 0.961i)3-s + (0.552 − 0.833i)4-s + (−0.153 − 0.988i)5-s + (0.213 + 0.976i)6-s + (0.816 − 0.577i)7-s + (0.0922 − 0.995i)8-s + (−0.850 − 0.526i)9-s + (−0.602 − 0.798i)10-s + (−0.0307 + 0.999i)11-s + (0.650 + 0.759i)12-s + (0.0922 + 0.995i)13-s + (0.445 − 0.895i)14-s + (0.992 + 0.122i)15-s + (−0.389 − 0.920i)16-s + (0.213 − 0.976i)17-s + ⋯ |
L(s) = 1 | + (0.881 − 0.473i)2-s + (−0.273 + 0.961i)3-s + (0.552 − 0.833i)4-s + (−0.153 − 0.988i)5-s + (0.213 + 0.976i)6-s + (0.816 − 0.577i)7-s + (0.0922 − 0.995i)8-s + (−0.850 − 0.526i)9-s + (−0.602 − 0.798i)10-s + (−0.0307 + 0.999i)11-s + (0.650 + 0.759i)12-s + (0.0922 + 0.995i)13-s + (0.445 − 0.895i)14-s + (0.992 + 0.122i)15-s + (−0.389 − 0.920i)16-s + (0.213 − 0.976i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.459510942 - 0.4780147119i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.459510942 - 0.4780147119i\) |
\(L(1)\) |
\(\approx\) |
\(1.478378701 - 0.3168895252i\) |
\(L(1)\) |
\(\approx\) |
\(1.478378701 - 0.3168895252i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 103 | \( 1 \) |
good | 2 | \( 1 + (0.881 - 0.473i)T \) |
| 3 | \( 1 + (-0.273 + 0.961i)T \) |
| 5 | \( 1 + (-0.153 - 0.988i)T \) |
| 7 | \( 1 + (0.816 - 0.577i)T \) |
| 11 | \( 1 + (-0.0307 + 0.999i)T \) |
| 13 | \( 1 + (0.0922 + 0.995i)T \) |
| 17 | \( 1 + (0.213 - 0.976i)T \) |
| 19 | \( 1 + (0.969 + 0.243i)T \) |
| 23 | \( 1 + (-0.850 + 0.526i)T \) |
| 29 | \( 1 + (-0.779 + 0.626i)T \) |
| 31 | \( 1 + (-0.602 + 0.798i)T \) |
| 37 | \( 1 + (-0.982 + 0.183i)T \) |
| 41 | \( 1 + (-0.153 + 0.988i)T \) |
| 43 | \( 1 + (0.650 - 0.759i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.696 + 0.717i)T \) |
| 59 | \( 1 + (0.816 + 0.577i)T \) |
| 61 | \( 1 + (0.739 + 0.673i)T \) |
| 67 | \( 1 + (-0.908 + 0.417i)T \) |
| 71 | \( 1 + (-0.779 - 0.626i)T \) |
| 73 | \( 1 + (0.932 - 0.361i)T \) |
| 79 | \( 1 + (0.932 + 0.361i)T \) |
| 83 | \( 1 + (-0.908 - 0.417i)T \) |
| 89 | \( 1 + (0.445 - 0.895i)T \) |
| 97 | \( 1 + (0.213 + 0.976i)T \) |
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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
−30.19518789103197815632409945363, −29.38562207887122667920265047837, −27.925804962771984032467939296354, −26.50272512086190714929055644518, −25.530306335395619403740131810771, −24.37850974930489481570909458775, −23.94623460664268002722081260387, −22.56766950989794257985542834847, −22.090799303062456203016957341588, −20.70823667063934252805101376600, −19.25829906302513942320083679403, −18.16973725431805997283937038612, −17.348993235972043706326108709491, −15.85662558857801937710258906838, −14.69711778777261183058444212792, −13.93331870615065642051886483498, −12.75021877180308680873261937055, −11.60490077742323852809573783073, −10.88715591178762123066036379019, −8.25576100248510348273446923781, −7.60172981297797625514411227386, −6.17225248642457052431565979402, −5.49213388650768617126103963080, −3.45353175157909318029477738497, −2.19690756501804821373587341865,
1.60384453237630171147665237900, 3.725517216334910114888077293027, 4.68628450271261491002733522433, 5.383513782050031818193893617802, 7.2838714742561263232885368053, 9.162397832462977359577512935050, 10.14048761514976736197195759561, 11.50221215585699905140167903808, 12.101963634988602382979706803473, 13.70172986815301601938298516656, 14.59463682583671149833868674443, 15.84926816759543391421984094751, 16.61522991003566601850461765828, 17.990032037899787010201778171497, 19.91586393435148219374729197863, 20.53550816819170686878975751700, 21.18471073233368607415588669653, 22.36114386155535518115943008069, 23.422568860314473809767816033381, 24.11845988754092346834207365964, 25.43021844127609572752358587681, 26.96361537505588867483158667409, 27.91366065007938847666110860435, 28.57888610759384022829772240118, 29.63549003065921328855222227704