L(s) = 1 | + (0.654 + 0.755i)3-s + (0.959 + 0.281i)5-s + (−0.959 − 0.281i)7-s + (−0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.654 − 0.755i)13-s + (0.415 + 0.909i)15-s + (0.415 − 0.909i)17-s + (0.142 − 0.989i)19-s + (−0.415 − 0.909i)21-s + (−0.142 + 0.989i)23-s + (0.841 + 0.540i)25-s + (−0.841 + 0.540i)27-s + (−0.959 − 0.281i)29-s + (−0.142 − 0.989i)31-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)3-s + (0.959 + 0.281i)5-s + (−0.959 − 0.281i)7-s + (−0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.654 − 0.755i)13-s + (0.415 + 0.909i)15-s + (0.415 − 0.909i)17-s + (0.142 − 0.989i)19-s + (−0.415 − 0.909i)21-s + (−0.142 + 0.989i)23-s + (0.841 + 0.540i)25-s + (−0.841 + 0.540i)27-s + (−0.959 − 0.281i)29-s + (−0.142 − 0.989i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.918436540 - 0.6158619103i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.918436540 - 0.6158619103i\) |
\(L(1)\) |
\(\approx\) |
\(1.245179206 + 0.1547971408i\) |
\(L(1)\) |
\(\approx\) |
\(1.245179206 + 0.1547971408i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (0.654 + 0.755i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (-0.959 - 0.281i)T \) |
| 11 | \( 1 + (-0.959 + 0.281i)T \) |
| 13 | \( 1 + (-0.654 - 0.755i)T \) |
| 17 | \( 1 + (0.415 - 0.909i)T \) |
| 19 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (-0.142 + 0.989i)T \) |
| 29 | \( 1 + (-0.959 - 0.281i)T \) |
| 31 | \( 1 + (-0.142 - 0.989i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.654 - 0.755i)T \) |
| 43 | \( 1 + (0.959 - 0.281i)T \) |
| 47 | \( 1 + (0.654 - 0.755i)T \) |
| 53 | \( 1 + (0.654 + 0.755i)T \) |
| 59 | \( 1 + (0.654 - 0.755i)T \) |
| 61 | \( 1 + (0.841 - 0.540i)T \) |
| 67 | \( 1 + (-0.654 - 0.755i)T \) |
| 71 | \( 1 + (0.959 - 0.281i)T \) |
| 73 | \( 1 + (-0.142 - 0.989i)T \) |
| 79 | \( 1 + (0.142 + 0.989i)T \) |
| 83 | \( 1 + (-0.415 + 0.909i)T \) |
| 97 | \( 1 + (-0.959 - 0.281i)T \) |
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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
−22.4362994338373443898581458578, −21.492797764627623789816018006444, −20.89904948210543344757737926461, −19.99024535649087541969646294083, −19.08006774293258120225226779851, −18.58000765322911285204915644228, −17.76054629114922108798155509191, −16.69129585491664031572808939923, −16.11307737460103642920907289287, −14.696917716198880263233660281001, −14.29986155378267102557442257680, −13.10761257236817755899591708338, −12.84374809804315264908661618193, −12.029044005715179384878298706621, −10.50885427842029971023405108737, −9.74535633933830006232013696513, −8.98831808983453942445716748973, −8.1299972346888945074607408754, −7.10362825985534561035436743247, −6.16133818801927331942250004372, −5.56409590967898885948616209980, −4.04980423250959418361237632211, −2.833180075397293774512298931677, −2.20340104220960994015416688342, −1.05721772914095747140393498574,
0.434066798691752445841768705619, 2.36773238920659034235330734656, 2.77332201724017122437230811994, 3.86443522439428011848072297450, 5.16832352938821732693049234618, 5.70387744369413122723300901060, 7.16807039158552742454895390259, 7.74992183523701822297262983573, 9.30195720750759157192199712082, 9.578737672573352769213696777229, 10.3404617917563249298925930953, 11.151883975839979448354511282957, 12.682791699478457613502406827953, 13.39270236960029839393748783630, 13.92848423988498228697083930936, 15.0794045729060990031555392823, 15.587390930594007518885648062712, 16.54546455792486104722712763637, 17.328462381658378414999003060255, 18.286863727927576072005074192727, 19.15701795500235886023518326018, 20.08725784351196706473924127128, 20.64919476573940286925311624014, 21.50686316056524111382038653330, 22.29696039978594835532455973388