L(s) = 1 | + (−1.19 + 0.760i)2-s + (0.707 + 0.707i)3-s + (0.844 − 1.81i)4-s + (0.432 + 2.19i)5-s + (−1.38 − 0.305i)6-s + (−0.611 + 0.611i)7-s + (0.371 + 2.80i)8-s + 1.00i·9-s + (−2.18 − 2.28i)10-s − 5.12i·11-s + (1.87 − 0.685i)12-s + (1.76 − 1.76i)13-s + (0.264 − 1.19i)14-s + (−1.24 + 1.85i)15-s + (−2.57 − 3.06i)16-s + (−3.76 − 3.76i)17-s + ⋯ |
L(s) = 1 | + (−0.843 + 0.537i)2-s + (0.408 + 0.408i)3-s + (0.422 − 0.906i)4-s + (0.193 + 0.981i)5-s + (−0.563 − 0.124i)6-s + (−0.231 + 0.231i)7-s + (0.131 + 0.991i)8-s + 0.333i·9-s + (−0.690 − 0.723i)10-s − 1.54i·11-s + (0.542 − 0.197i)12-s + (0.488 − 0.488i)13-s + (0.0706 − 0.319i)14-s + (−0.321 + 0.479i)15-s + (−0.643 − 0.765i)16-s + (−0.912 − 0.912i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.593269 + 0.362829i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.593269 + 0.362829i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.19 - 0.760i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.432 - 2.19i)T \) |
good | 7 | \( 1 + (0.611 - 0.611i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.12iT - 11T^{2} \) |
| 13 | \( 1 + (-1.76 + 1.76i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.76 + 3.76i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.22T + 19T^{2} \) |
| 23 | \( 1 + (-1.07 - 1.07i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.864iT - 29T^{2} \) |
| 31 | \( 1 - 7.81iT - 31T^{2} \) |
| 37 | \( 1 + (1.76 + 1.76i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.52T + 41T^{2} \) |
| 43 | \( 1 + (6.20 + 6.20i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.29 - 2.29i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.62 - 2.62i)T - 53iT^{2} \) |
| 59 | \( 1 - 0.528T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 + (-6.20 + 6.20i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.10iT - 71T^{2} \) |
| 73 | \( 1 + (2.25 - 2.25i)T - 73iT^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + (-7.95 - 7.95i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.25iT - 89T^{2} \) |
| 97 | \( 1 + (-0.793 - 0.793i)T + 97iT^{2} \) |
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
−15.60832160100677031397348988730, −14.33737008950654352740652010545, −13.61514662192035021469543101376, −11.32696421580510847118164097777, −10.61522855440015706435788908581, −9.352777320142061324897566286716, −8.336848530553082208476513259169, −6.89629736342839008472931529207, −5.64597234276793293522754012365, −2.99794960363558667981740078565,
1.87502238686416301347343536366, 4.22186681257837288851574594309, 6.69482606699020307559995109714, 8.026410503161687663804125371656, 9.110131533073862380999070467587, 10.01227946914180345037072447335, 11.58217129438887475327560269800, 12.72877450288727286117313601928, 13.28872046499776772258780600760, 15.07280915437925742761061930203