L(s) = 1 | + (0.587 + 0.809i)9-s + (1.76 + 0.278i)13-s + (−0.896 − 1.76i)17-s + (1.11 − 0.363i)29-s + (0.0489 − 0.309i)37-s + (−1.53 + 1.11i)41-s + i·49-s + (0.412 − 0.809i)53-s + (1.53 + 1.11i)61-s + (0.142 + 0.896i)73-s + (−0.309 + 0.951i)81-s + (0.690 − 0.951i)89-s + (−0.278 − 0.142i)97-s + 1.61·101-s + (−0.363 − 0.5i)109-s + ⋯ |
L(s) = 1 | + (0.587 + 0.809i)9-s + (1.76 + 0.278i)13-s + (−0.896 − 1.76i)17-s + (1.11 − 0.363i)29-s + (0.0489 − 0.309i)37-s + (−1.53 + 1.11i)41-s + i·49-s + (0.412 − 0.809i)53-s + (1.53 + 1.11i)61-s + (0.142 + 0.896i)73-s + (−0.309 + 0.951i)81-s + (0.690 − 0.951i)89-s + (−0.278 − 0.142i)97-s + 1.61·101-s + (−0.363 − 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.465384419\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.465384419\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.0489 + 0.309i)T + (-0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (-0.412 + 0.809i)T + (-0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 89 | \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2} \) |
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
−8.572826510963427494698232511226, −8.024996762893183696577410326444, −7.02519498708550208598214563735, −6.61958689752670733596629044534, −5.64562952951277556262959605093, −4.79155203502633540688866692971, −4.20064040665824189601359278718, −3.15585883393548520857470050963, −2.21647296787341595526501109389, −1.11519974014613452199786838663,
1.09943522073885087795436848309, 2.02921698656194227111275542751, 3.52244590849793936171477582073, 3.78078493873032308245740174740, 4.78554773091469377040526984307, 5.85173660139586633968019129454, 6.45012110743747835061881979040, 6.91291594940670005380097965151, 8.140434587313513454323523124661, 8.566934501482305945741206327626