| L(s) = 1 | + 3-s + 9-s − 2·11-s − 4·13-s + 2·17-s + 2·23-s + 27-s + 6·29-s + 4·31-s − 2·33-s + 10·37-s − 4·39-s + 2·41-s + 4·43-s + 8·47-s + 2·51-s − 10·53-s + 4·59-s − 8·61-s + 8·67-s + 2·69-s − 6·71-s + 4·73-s − 4·79-s + 81-s − 4·83-s + 6·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.485·17-s + 0.417·23-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.348·33-s + 1.64·37-s − 0.640·39-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 0.280·51-s − 1.37·53-s + 0.520·59-s − 1.02·61-s + 0.977·67-s + 0.240·69-s − 0.712·71-s + 0.468·73-s − 0.450·79-s + 1/9·81-s − 0.439·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.865535383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.865535383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.30085853291275, −13.96643510263319, −13.29351818932954, −12.87040986644984, −12.30263737341042, −11.99775012472145, −11.22017503043803, −10.70982991653176, −10.16612855442228, −9.643272498419478, −9.336240646548879, −8.541295030496020, −8.113572373708734, −7.527407735651244, −7.240391486348034, −6.407700898407460, −5.923089873841759, −5.130778431030683, −4.665049427348490, −4.147859748301046, −3.275301587847227, −2.650264471162388, −2.395266230781973, −1.321637736179472, −0.5814864070986525,
0.5814864070986525, 1.321637736179472, 2.395266230781973, 2.650264471162388, 3.275301587847227, 4.147859748301046, 4.665049427348490, 5.130778431030683, 5.923089873841759, 6.407700898407460, 7.240391486348034, 7.527407735651244, 8.113572373708734, 8.541295030496020, 9.336240646548879, 9.643272498419478, 10.16612855442228, 10.70982991653176, 11.22017503043803, 11.99775012472145, 12.30263737341042, 12.87040986644984, 13.29351818932954, 13.96643510263319, 14.30085853291275
