L(s) = 1 | − 3·5-s + 3·11-s − 2·13-s − 6·17-s + 2·19-s + 6·23-s + 4·25-s + 9·29-s − 7·31-s − 10·37-s + 4·43-s + 12·47-s − 3·53-s − 9·55-s − 3·59-s + 4·61-s + 6·65-s − 2·67-s − 2·73-s − 5·79-s + 9·83-s + 18·85-s + 6·89-s − 6·95-s + 13·97-s − 6·101-s − 16·103-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.904·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s + 1.25·23-s + 4/5·25-s + 1.67·29-s − 1.25·31-s − 1.64·37-s + 0.609·43-s + 1.75·47-s − 0.412·53-s − 1.21·55-s − 0.390·59-s + 0.512·61-s + 0.744·65-s − 0.244·67-s − 0.234·73-s − 0.562·79-s + 0.987·83-s + 1.95·85-s + 0.635·89-s − 0.615·95-s + 1.31·97-s − 0.597·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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
−7.42889004200935283095691888550, −7.03499149561317652197237395885, −6.43279889442694837895749028653, −5.30774285622288732019904751538, −4.60628839537154430808649782406, −4.00360016328830772086960715028, −3.28885916258822933348831841519, −2.36884465130644791006072860892, −1.12153207349611637065994537186, 0,
1.12153207349611637065994537186, 2.36884465130644791006072860892, 3.28885916258822933348831841519, 4.00360016328830772086960715028, 4.60628839537154430808649782406, 5.30774285622288732019904751538, 6.43279889442694837895749028653, 7.03499149561317652197237395885, 7.42889004200935283095691888550