| L(s) = 1 | + 1.73·5-s + 1.73·7-s − 3·11-s + 4·19-s − 6.92·23-s − 2.00·25-s − 6.92·29-s + 1.73·31-s + 2.99·35-s − 6.92·37-s − 12·41-s + 4·43-s + 6.92·47-s − 4·49-s + 8.66·53-s − 5.19·55-s − 13.8·61-s − 4·67-s − 13.8·71-s + 7·73-s − 5.19·77-s − 10.3·79-s + 9·83-s − 12·89-s + 6.92·95-s + 7·97-s + 8.66·101-s + ⋯ |
| L(s) = 1 | + 0.774·5-s + 0.654·7-s − 0.904·11-s + 0.917·19-s − 1.44·23-s − 0.400·25-s − 1.28·29-s + 0.311·31-s + 0.507·35-s − 1.13·37-s − 1.87·41-s + 0.609·43-s + 1.01·47-s − 0.571·49-s + 1.18·53-s − 0.700·55-s − 1.77·61-s − 0.488·67-s − 1.64·71-s + 0.819·73-s − 0.592·77-s − 1.16·79-s + 0.987·83-s − 1.27·89-s + 0.710·95-s + 0.710·97-s + 0.861·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
| 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
−7.65247107392447576789109877036, −7.02433772523670047351469195779, −5.95356055282443509990321990097, −5.57415191429352364280981728979, −4.90602093412832789914788215331, −4.00030666411117668994217207009, −3.09268833059851629044549522701, −2.11819607768010710816394026456, −1.53705536348298108439814353314, 0,
1.53705536348298108439814353314, 2.11819607768010710816394026456, 3.09268833059851629044549522701, 4.00030666411117668994217207009, 4.90602093412832789914788215331, 5.57415191429352364280981728979, 5.95356055282443509990321990097, 7.02433772523670047351469195779, 7.65247107392447576789109877036
