L(s) = 1 | + (−1.16 − 0.801i)2-s + (−1.70 − 0.286i)3-s + (0.715 + 1.86i)4-s + (1.09 − 1.94i)5-s + (1.76 + 1.70i)6-s + (−0.0500 − 0.186i)7-s + (0.662 − 2.74i)8-s + (2.83 + 0.979i)9-s + (−2.84 + 1.39i)10-s + (−4.51 − 2.60i)11-s + (−0.687 − 3.39i)12-s + (−6.23 − 1.67i)13-s + (−0.0913 + 0.257i)14-s + (−2.43 + 3.01i)15-s + (−2.97 + 2.67i)16-s + (−0.305 − 0.305i)17-s + ⋯ |
L(s) = 1 | + (−0.823 − 0.566i)2-s + (−0.986 − 0.165i)3-s + (0.357 + 0.933i)4-s + (0.491 − 0.871i)5-s + (0.718 + 0.695i)6-s + (−0.0189 − 0.0706i)7-s + (0.234 − 0.972i)8-s + (0.945 + 0.326i)9-s + (−0.898 + 0.439i)10-s + (−1.36 − 0.785i)11-s + (−0.198 − 0.980i)12-s + (−1.72 − 0.463i)13-s + (−0.0244 + 0.0689i)14-s + (−0.628 + 0.777i)15-s + (−0.743 + 0.668i)16-s + (−0.0740 − 0.0740i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0850625 - 0.396408i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0850625 - 0.396408i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.16 + 0.801i)T \) |
| 3 | \( 1 + (1.70 + 0.286i)T \) |
| 5 | \( 1 + (-1.09 + 1.94i)T \) |
good | 7 | \( 1 + (0.0500 + 0.186i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (4.51 + 2.60i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (6.23 + 1.67i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (0.305 + 0.305i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 23 | \( 1 + (-1.13 + 4.24i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (2.28 + 1.31i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.124 + 0.0717i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.83 - 2.83i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.76 + 3.05i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.96 + 2.13i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-2.07 - 7.73i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (3.86 - 3.86i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.587 + 1.01i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.54 + 4.40i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.12 - 2.44i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 6.80iT - 71T^{2} \) |
| 73 | \( 1 + (1.96 - 1.96i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.89 + 3.28i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.57 - 2.03i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 13.0iT - 89T^{2} \) |
| 97 | \( 1 + (5.44 - 1.45i)T + (84.0 - 48.5i)T^{2} \) |
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
−12.32473489818838904464364018256, −11.11281103505066708203921666346, −10.26455184476084460355972699575, −9.512069534834731320216801448117, −8.157737600172002121422643577627, −7.27646387184152414368463130795, −5.72417298567498088800473937136, −4.69894171819607409756063038257, −2.48544334518533961458149402244, −0.51072981937686292872288853537,
2.28405079202752012412731446617, 4.93749995041233608617507766930, 5.75521690044627165354768286167, 7.12652223651323707335058148135, 7.47785918918586381750826905547, 9.513330201231952186426306856321, 9.994360504063092146640369192226, 10.83997690454648993030443732091, 11.80059640280094798134224033690, 13.01264253996746467797277852617