L(s) = 1 | + (0.707 − 0.707i)3-s + (−1.98 + 1.02i)5-s + (−1.54 + 1.54i)7-s − 1.00i·9-s − 1.73i·11-s + (2.14 − 2.14i)13-s + (−0.676 + 2.13i)15-s + (−0.622 + 0.622i)17-s + 1.52·19-s + 2.17i·21-s + (4.73 + 0.781i)23-s + (2.88 − 4.08i)25-s + (−0.707 − 0.707i)27-s + 8.11i·29-s + 3.88·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.887 + 0.460i)5-s + (−0.582 + 0.582i)7-s − 0.333i·9-s − 0.524i·11-s + (0.595 − 0.595i)13-s + (−0.174 + 0.550i)15-s + (−0.150 + 0.150i)17-s + 0.348·19-s + 0.475i·21-s + (0.986 + 0.162i)23-s + (0.576 − 0.817i)25-s + (−0.136 − 0.136i)27-s + 1.50i·29-s + 0.697·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.537873593\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.537873593\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.98 - 1.02i)T \) |
| 23 | \( 1 + (-4.73 - 0.781i)T \) |
good | 7 | \( 1 + (1.54 - 1.54i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.73iT - 11T^{2} \) |
| 13 | \( 1 + (-2.14 + 2.14i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.622 - 0.622i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.52T + 19T^{2} \) |
| 29 | \( 1 - 8.11iT - 29T^{2} \) |
| 31 | \( 1 - 3.88T + 31T^{2} \) |
| 37 | \( 1 + (-4.35 + 4.35i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.64T + 41T^{2} \) |
| 43 | \( 1 + (-5.05 - 5.05i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.65 - 2.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.53 - 6.53i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.33iT - 59T^{2} \) |
| 61 | \( 1 - 3.35iT - 61T^{2} \) |
| 67 | \( 1 + (-0.825 + 0.825i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.10T + 71T^{2} \) |
| 73 | \( 1 + (-7.83 + 7.83i)T - 73iT^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + (10.6 + 10.6i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 + (-1.44 + 1.44i)T - 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.302757240746461872286754576030, −8.800822781289801147429384143027, −7.926617372385971723803797619406, −7.27040594024625266852133611934, −6.37150462831247605948790942332, −5.62092112016365488321512635934, −4.29546456767999416200234880756, −3.22334212925051951942833601433, −2.76116858982389941039358960081, −0.953709434029377079813001614453,
0.843021706136556049973398972114, 2.53837389151592945455540732473, 3.73692688153160824962820682662, 4.23247044820294972368726659932, 5.15514931925415077668962804364, 6.44923240137138340238834535932, 7.23316221878862045692654259358, 8.001182623189675653317880022293, 8.809006285131110608441055967645, 9.515452571702371647073574184755