L(s) = 1 | + (−1.14 − 0.828i)2-s + 0.692i·3-s + (0.627 + 1.89i)4-s + (2.22 − 0.245i)5-s + (0.573 − 0.794i)6-s + (−0.343 + 0.343i)7-s + (0.853 − 2.69i)8-s + 2.52·9-s + (−2.75 − 1.55i)10-s + (0.843 − 0.843i)11-s + (−1.31 + 0.434i)12-s − 3.68·13-s + (0.678 − 0.109i)14-s + (0.169 + 1.53i)15-s + (−3.21 + 2.38i)16-s + (0.412 − 0.412i)17-s + ⋯ |
L(s) = 1 | + (−0.810 − 0.585i)2-s + 0.399i·3-s + (0.313 + 0.949i)4-s + (0.993 − 0.109i)5-s + (0.234 − 0.324i)6-s + (−0.129 + 0.129i)7-s + (0.301 − 0.953i)8-s + 0.840·9-s + (−0.869 − 0.493i)10-s + (0.254 − 0.254i)11-s + (−0.379 + 0.125i)12-s − 1.02·13-s + (0.181 − 0.0292i)14-s + (0.0438 + 0.397i)15-s + (−0.802 + 0.596i)16-s + (0.0999 − 0.0999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.760534 - 0.0753991i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.760534 - 0.0753991i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.14 + 0.828i)T \) |
| 5 | \( 1 + (-2.22 + 0.245i)T \) |
good | 3 | \( 1 - 0.692iT - 3T^{2} \) |
| 7 | \( 1 + (0.343 - 0.343i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.843 + 0.843i)T - 11iT^{2} \) |
| 13 | \( 1 + 3.68T + 13T^{2} \) |
| 17 | \( 1 + (-0.412 + 0.412i)T - 17iT^{2} \) |
| 19 | \( 1 + (5.37 - 5.37i)T - 19iT^{2} \) |
| 23 | \( 1 + (3.08 + 3.08i)T + 23iT^{2} \) |
| 29 | \( 1 + (4.22 + 4.22i)T + 29iT^{2} \) |
| 31 | \( 1 + 8.75iT - 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 + 2.54iT - 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 + (-4.56 - 4.56i)T + 47iT^{2} \) |
| 53 | \( 1 - 6.07iT - 53T^{2} \) |
| 59 | \( 1 + (7.33 + 7.33i)T + 59iT^{2} \) |
| 61 | \( 1 + (4.81 - 4.81i)T - 61iT^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 2.97T + 71T^{2} \) |
| 73 | \( 1 + (6.87 - 6.87i)T - 73iT^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.15iT - 83T^{2} \) |
| 89 | \( 1 - 1.10T + 89T^{2} \) |
| 97 | \( 1 + (-7.15 + 7.15i)T - 97iT^{2} \) |
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
−14.32911196938981672369923699334, −12.96396490285967532589055366289, −12.22941768190475132503091981162, −10.67484925908879209136033621177, −9.920036285460541616597579202672, −9.142674657050518975885936962878, −7.68575851574117954396187097044, −6.17392987682462653714684503891, −4.19306731332274611733261548386, −2.14103448986340814153092380725,
1.94286648681342546356333279966, 5.04345819430678989282960683638, 6.57390776362773085668724857712, 7.28048831174309041384774241786, 8.887365211808021751229348529468, 9.862236195444915600025779205066, 10.71243111293217889428016139248, 12.39084965769282036065335833702, 13.51069269286602401990485145169, 14.55035488780389464936975267720