L(s) = 1 | + (−0.951 − 0.309i)9-s + (0.278 − 0.142i)13-s + (1.76 − 0.278i)17-s + (−1.11 − 1.53i)29-s + (0.412 + 0.809i)37-s + (0.363 − 1.11i)41-s + i·49-s + (1.95 + 0.309i)53-s + (−0.363 − 1.11i)61-s + (0.896 − 1.76i)73-s + (0.809 + 0.587i)81-s + (1.80 − 0.587i)89-s + (0.142 − 0.896i)97-s − 0.618·101-s + (−1.53 − 0.5i)109-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)9-s + (0.278 − 0.142i)13-s + (1.76 − 0.278i)17-s + (−1.11 − 1.53i)29-s + (0.412 + 0.809i)37-s + (0.363 − 1.11i)41-s + i·49-s + (1.95 + 0.309i)53-s + (−0.363 − 1.11i)61-s + (0.896 − 1.76i)73-s + (0.809 + 0.587i)81-s + (1.80 − 0.587i)89-s + (0.142 − 0.896i)97-s − 0.618·101-s + (−1.53 − 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.219460150\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.219460150\) |
\(L(1)\) |
|
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 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.412 - 0.809i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (-1.95 - 0.309i)T + (0.951 + 0.309i)T^{2} \) |
| 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 | \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
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
−8.478885472324683250174087789972, −7.84057033884969017682353192431, −7.24261362683337572403111034875, −6.02969586048932757912666489954, −5.81571222009704030093654407645, −4.88313122617080892124306144471, −3.79140280183946299740506364236, −3.16506333841467406646810723115, −2.18089425945419952803428963485, −0.796156898929570856522469151921,
1.17462569582701276217779496759, 2.37278246452391450312378259737, 3.31183418884635302665518492425, 3.97119494904179707700201623494, 5.31388922453337902750425552965, 5.50354317954882683202894764082, 6.45531679637615261693075260807, 7.36794006423324002193215077861, 7.964543406605789324298236769241, 8.685562029224743100475070326393