L(s) = 1 | + (0.479 + 0.877i)3-s + (−0.909 + 0.415i)5-s + (−0.349 + 0.936i)7-s + (−0.540 + 0.841i)9-s + (0.415 − 0.909i)11-s + (0.877 − 0.479i)13-s + (−0.800 − 0.599i)15-s + (0.989 + 0.142i)17-s + (0.977 − 0.212i)19-s + (−0.989 + 0.142i)21-s + (0.212 + 0.977i)23-s + (0.654 − 0.755i)25-s + (−0.997 − 0.0713i)27-s + (0.936 + 0.349i)29-s + (0.212 − 0.977i)31-s + ⋯ |
L(s) = 1 | + (0.479 + 0.877i)3-s + (−0.909 + 0.415i)5-s + (−0.349 + 0.936i)7-s + (−0.540 + 0.841i)9-s + (0.415 − 0.909i)11-s + (0.877 − 0.479i)13-s + (−0.800 − 0.599i)15-s + (0.989 + 0.142i)17-s + (0.977 − 0.212i)19-s + (−0.989 + 0.142i)21-s + (0.212 + 0.977i)23-s + (0.654 − 0.755i)25-s + (−0.997 − 0.0713i)27-s + (0.936 + 0.349i)29-s + (0.212 − 0.977i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0173 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0173 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.683972623 + 1.655071263i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.683972623 + 1.655071263i\) |
\(L(1)\) |
\(\approx\) |
\(1.115616769 + 0.5266962742i\) |
\(L(1)\) |
\(\approx\) |
\(1.115616769 + 0.5266962742i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (0.479 + 0.877i)T \) |
| 5 | \( 1 + (-0.909 + 0.415i)T \) |
| 7 | \( 1 + (-0.349 + 0.936i)T \) |
| 11 | \( 1 + (0.415 - 0.909i)T \) |
| 13 | \( 1 + (0.877 - 0.479i)T \) |
| 17 | \( 1 + (0.989 + 0.142i)T \) |
| 19 | \( 1 + (0.977 - 0.212i)T \) |
| 23 | \( 1 + (0.212 + 0.977i)T \) |
| 29 | \( 1 + (0.936 + 0.349i)T \) |
| 31 | \( 1 + (0.212 - 0.977i)T \) |
| 37 | \( 1 + (0.707 - 0.707i)T \) |
| 41 | \( 1 + (0.877 + 0.479i)T \) |
| 43 | \( 1 + (0.936 - 0.349i)T \) |
| 47 | \( 1 + (0.281 + 0.959i)T \) |
| 53 | \( 1 + (0.281 - 0.959i)T \) |
| 59 | \( 1 + (-0.479 + 0.877i)T \) |
| 61 | \( 1 + (0.0713 - 0.997i)T \) |
| 67 | \( 1 + (0.959 + 0.281i)T \) |
| 71 | \( 1 + (-0.909 - 0.415i)T \) |
| 73 | \( 1 + (-0.841 + 0.540i)T \) |
| 79 | \( 1 + (0.540 + 0.841i)T \) |
| 83 | \( 1 + (-0.800 + 0.599i)T \) |
| 97 | \( 1 + (0.415 + 0.909i)T \) |
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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
−22.61541906197575059590373728804, −20.96852295614007792195778551446, −20.393452129005590531285755877086, −19.801672568196436015931339905901, −19.05189472750597142449085166839, −18.29705965724349395314384553976, −17.30939022518848042385618671119, −16.457178952880884749985789406115, −15.71485887913794465186161387211, −14.54208896197279757108165445719, −13.97843397850977515937319115083, −13.02109446365387433973979371372, −12.242790302458578061238404174886, −11.67367689313483768772715054835, −10.45892742068131824212928832985, −9.39858921436732269207797410857, −8.51511474627406377661354366156, −7.608167862285715022939244974027, −7.07643911750383029162459564827, −6.13686637039567932343590117215, −4.62447965320576590427080280437, −3.79447804084658248626979529303, −2.91142926669373013968861614887, −1.32740984619624617751966295821, −0.75983838070045639570627469665,
0.892122273929827771671765814612, 2.758111220143231105844161547237, 3.30075966221300356119957432120, 4.08191982024030519281816508938, 5.42280587964336320836530353440, 6.08449370853985011364478614414, 7.56869887430029504429484280678, 8.28434832547530874606155120882, 9.0920235968200993226714228474, 9.90861688818731615167183353974, 11.07058862008947847130444824886, 11.49189163369894507845136451393, 12.56950001141340174491113817282, 13.744658957831668013391843631108, 14.50125865746780764741213070724, 15.349568674112386299646035627065, 15.95552861222741758334760151114, 16.42547005204604385651091722016, 17.807323771890353277040026720667, 18.83605663979572332922525489667, 19.31569687906358017481861985949, 20.108286294319729085439685547558, 21.09728831556670464544005529178, 21.75308271518442412033553799183, 22.52628734033375632597507548860