L(s) = 1 | + (1.37 + 0.319i)2-s + (1.79 + 0.879i)4-s + (−1.42 + 1.72i)5-s + (3.11 − 3.11i)7-s + (2.19 + 1.78i)8-s + (−2.51 + 1.91i)10-s + 1.17·11-s + (−2.15 + 2.15i)13-s + (5.29 − 3.30i)14-s + (2.45 + 3.15i)16-s + (1.33 + 1.33i)17-s − 0.322·19-s + (−4.07 + 1.84i)20-s + (1.62 + 0.375i)22-s + (−4.71 + 4.71i)23-s + ⋯ |
L(s) = 1 | + (0.974 + 0.225i)2-s + (0.898 + 0.439i)4-s + (−0.637 + 0.770i)5-s + (1.17 − 1.17i)7-s + (0.775 + 0.630i)8-s + (−0.795 + 0.606i)10-s + 0.355·11-s + (−0.596 + 0.596i)13-s + (1.41 − 0.882i)14-s + (0.613 + 0.789i)16-s + (0.323 + 0.323i)17-s − 0.0739·19-s + (−0.911 + 0.411i)20-s + (0.346 + 0.0801i)22-s + (−0.982 + 0.982i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.29190 + 0.706692i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.29190 + 0.706692i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 - 0.319i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.42 - 1.72i)T \) |
good | 7 | \( 1 + (-3.11 + 3.11i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 + (2.15 - 2.15i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.33 - 1.33i)T + 17iT^{2} \) |
| 19 | \( 1 + 0.322T + 19T^{2} \) |
| 23 | \( 1 + (4.71 - 4.71i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.63iT - 29T^{2} \) |
| 31 | \( 1 + 0.0675T + 31T^{2} \) |
| 37 | \( 1 + (7.60 + 7.60i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.19iT - 41T^{2} \) |
| 43 | \( 1 + (-6.70 + 6.70i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.34 + 7.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.73 + 5.73i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.68iT - 59T^{2} \) |
| 61 | \( 1 - 12.5iT - 61T^{2} \) |
| 67 | \( 1 + (1.87 + 1.87i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.18iT - 71T^{2} \) |
| 73 | \( 1 + (-3.97 - 3.97i)T + 73iT^{2} \) |
| 79 | \( 1 - 9.66iT - 79T^{2} \) |
| 83 | \( 1 + (-0.585 - 0.585i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.557T + 89T^{2} \) |
| 97 | \( 1 + (10.5 - 10.5i)T - 97iT^{2} \) |
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
−11.62271873493519726072929768076, −10.90204692074357578130686159945, −10.07565445109469093101507628982, −8.277697802642588878166592213864, −7.45844789831406383902381839870, −6.93975281967422160582864263873, −5.58435790798571019763557005890, −4.26862290249492515631875374053, −3.76561244299867539166445240742, −2.01192538055345542726572011914,
1.65369428608771081521267895679, 3.10494596779561324815423156999, 4.65277547994584241094379421986, 5.08087560288095423956996062709, 6.23523245302148857826013573669, 7.68604187997170889466556958696, 8.380436881673100926788727860400, 9.537608779115760383303076360560, 10.83860664162682893788697822797, 11.64407972798087263335288752535