L(s) = 1 | + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (−0.974 + 0.222i)8-s + (0.0747 + 0.997i)11-s + (0.997 − 0.0747i)13-s + (−0.222 + 0.974i)16-s + (−0.930 + 0.365i)17-s + (0.5 + 0.866i)19-s + (0.930 + 0.365i)22-s + (0.149 + 0.988i)23-s + (0.365 − 0.930i)26-s + (−0.365 − 0.930i)29-s + 31-s + (0.781 + 0.623i)32-s + (−0.0747 + 0.997i)34-s + ⋯ |
L(s) = 1 | + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (−0.974 + 0.222i)8-s + (0.0747 + 0.997i)11-s + (0.997 − 0.0747i)13-s + (−0.222 + 0.974i)16-s + (−0.930 + 0.365i)17-s + (0.5 + 0.866i)19-s + (0.930 + 0.365i)22-s + (0.149 + 0.988i)23-s + (0.365 − 0.930i)26-s + (−0.365 − 0.930i)29-s + 31-s + (0.781 + 0.623i)32-s + (−0.0747 + 0.997i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3544444666 + 0.4500484017i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3544444666 + 0.4500484017i\) |
\(L(1)\) |
\(\approx\) |
\(1.021548043 - 0.3843065867i\) |
\(L(1)\) |
\(\approx\) |
\(1.021548043 - 0.3843065867i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.433 - 0.900i)T \) |
| 11 | \( 1 + (0.0747 + 0.997i)T \) |
| 13 | \( 1 + (0.997 - 0.0747i)T \) |
| 17 | \( 1 + (-0.930 + 0.365i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.149 + 0.988i)T \) |
| 29 | \( 1 + (-0.365 - 0.930i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.149 - 0.988i)T \) |
| 41 | \( 1 + (-0.733 + 0.680i)T \) |
| 43 | \( 1 + (-0.680 + 0.733i)T \) |
| 47 | \( 1 + (0.433 - 0.900i)T \) |
| 53 | \( 1 + (0.149 + 0.988i)T \) |
| 59 | \( 1 + (0.222 - 0.974i)T \) |
| 61 | \( 1 + (0.623 - 0.781i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.623 + 0.781i)T \) |
| 73 | \( 1 + (-0.997 - 0.0747i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (-0.997 - 0.0747i)T \) |
| 89 | \( 1 + (-0.826 - 0.563i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−19.14080490331739183158081400842, −18.43174798754152609712851798582, −17.880238090804933309984877171115, −17.00075722920387290984740125014, −16.34443311844499158557420997393, −15.744843969032454612706699529598, −15.144584151377444279563900720396, −14.192696722223417228280440982901, −13.53298711153255431025276328152, −13.20887243213371709974818529595, −12.104717895092382045726999246323, −11.38805419100987488336179951193, −10.63765939594487471433380828772, −9.466472050269180093061146629996, −8.52426569064143071774397690993, −8.4811394667752836758064829627, −7.11049840060461268391068554008, −6.63404635139878290367983616970, −5.83807147820049076841318945913, −5.02437385699258069928053224754, −4.24680620560792674991594347930, −3.344016191298891308559364673365, −2.63339169746469900613363294770, −1.08205631501091246945723648795, −0.08975341964190851325191132004,
1.218395520952162884444628060157, 1.85634274051498104559062222164, 2.81972374965183206090569446586, 3.820050345993039812119978689002, 4.30640479488614894724082846106, 5.327278635958443580428243286633, 6.04065327752425865518709062743, 6.90706649359017635924144194673, 8.030434353401607578372907670340, 8.79235157995864191015088642838, 9.74059436550478349075999652797, 10.12874242464561168286573983816, 11.19778012465352529388125688495, 11.60283841680354533936058880751, 12.49957528124106906409753819050, 13.186275656336609022130808534776, 13.72877949352121745591920847405, 14.60063080981910494645269277506, 15.346267091741225798091395861685, 15.87204472206622650087567336933, 17.153446673221375239561320452974, 17.74400890926192396170369844569, 18.461196881819888978941625320900, 19.09808676639383363603521901873, 20.03935321155380006363286763534