L(s) = 1 | − i·3-s − 9-s − 4·11-s + 2i·13-s + 2i·17-s + 8·19-s − 4i·23-s + i·27-s + 6·29-s + 4i·33-s − 2i·37-s + 2·39-s − 6·41-s − 4i·43-s − 12i·47-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.333·9-s − 1.20·11-s + 0.554i·13-s + 0.485i·17-s + 1.83·19-s − 0.834i·23-s + 0.192i·27-s + 1.11·29-s + 0.696i·33-s − 0.328i·37-s + 0.320·39-s − 0.937·41-s − 0.609i·43-s − 1.75i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.559654199\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.559654199\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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
−8.529389122583693293741640865042, −8.190393920947441415700813720900, −7.15806364265476234137745060548, −6.75148687208535328069321928687, −5.55339513595594029290523741562, −5.12042229147671765131384252835, −3.89059680202904436367062816236, −2.87783097938211242420526184109, −1.99851166935410544626595899634, −0.65018523303646176530392063763,
1.00217395972657893711427271711, 2.68003945870910668634102494478, 3.20338781620179052333488351100, 4.36816029174412791771296539175, 5.32840170322913764121323210682, 5.58966397014236965753719663333, 6.90459188054174155988711236385, 7.66662785429589770897642491505, 8.270377436592026487822030625073, 9.201145566901437476918609800502