L(s) = 1 | + (−0.156 − 0.987i)5-s + (0.896 + 1.76i)13-s + (0.610 − 0.0966i)17-s + (−0.951 + 0.309i)25-s + (0.734 − 0.533i)29-s + (0.809 − 0.412i)37-s + (1.87 + 0.610i)41-s + i·49-s + (−1.59 − 0.253i)53-s + (−0.363 − 1.11i)61-s + (1.59 − 1.16i)65-s + (0.278 + 0.142i)73-s + (−0.190 − 0.587i)85-s + (−0.550 − 1.69i)89-s + (1.76 + 0.278i)97-s + ⋯ |
L(s) = 1 | + (−0.156 − 0.987i)5-s + (0.896 + 1.76i)13-s + (0.610 − 0.0966i)17-s + (−0.951 + 0.309i)25-s + (0.734 − 0.533i)29-s + (0.809 − 0.412i)37-s + (1.87 + 0.610i)41-s + i·49-s + (−1.59 − 0.253i)53-s + (−0.363 − 1.11i)61-s + (1.59 − 1.16i)65-s + (0.278 + 0.142i)73-s + (−0.190 − 0.587i)85-s + (−0.550 − 1.69i)89-s + (1.76 + 0.278i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.328520601\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.328520601\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.156 + 0.987i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.610 + 0.0966i)T + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (-0.734 + 0.533i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.412i)T + (0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (-1.87 - 0.610i)T + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (1.59 + 0.253i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2} \) |
| 79 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + (0.550 + 1.69i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \) |
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
−8.725615539769453779287074724004, −8.042705549381393225085205833790, −7.33840057420360788628017146974, −6.29085338100211500341647259895, −5.85573819791277670292802779839, −4.60213617538896462675438147560, −4.35101025796294310236399061414, −3.29692793893087266062611180122, −2.01004254692297942261471047378, −1.07578613619746868521631991607,
1.04346522296761098263226343852, 2.54111789546927266121644305788, 3.22070817995563944591413679833, 3.92085286254233371845352983621, 5.08809134614581651086690728098, 5.92872278955977004202333603488, 6.40108964217679267426320692948, 7.47120467722528340817611584339, 7.87865355469595488223412369863, 8.620059380890689167379639811480