L(s) = 1 | + (0.623 − 1.07i)2-s + (−0.277 − 0.480i)4-s + (0.5 + 0.866i)7-s + 0.554·8-s + (−0.5 − 0.866i)9-s + (−0.222 + 0.385i)11-s + 1.24·14-s + (0.623 − 1.07i)16-s − 1.24·18-s + (0.277 + 0.480i)22-s + (0.900 − 1.56i)23-s + 25-s + (0.277 − 0.480i)28-s + (−0.623 + 1.07i)29-s + (−0.500 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.623 − 1.07i)2-s + (−0.277 − 0.480i)4-s + (0.5 + 0.866i)7-s + 0.554·8-s + (−0.5 − 0.866i)9-s + (−0.222 + 0.385i)11-s + 1.24·14-s + (0.623 − 1.07i)16-s − 1.24·18-s + (0.277 + 0.480i)22-s + (0.900 − 1.56i)23-s + 25-s + (0.277 − 0.480i)28-s + (−0.623 + 1.07i)29-s + (−0.500 − 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.571067341\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.571067341\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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
−10.07376879211944631861308762758, −8.960114159933826164811645700082, −8.500907165319582396464077953092, −7.26735557627726616003749316830, −6.36467442411643166666618709920, −5.16593464651575966625470920477, −4.64709120990023490782446981041, −3.32909495387190627498577820340, −2.71478510887405314283584992693, −1.52120211941783746133189916310,
1.64286039813160658046212657353, 3.25470440806257884946507219775, 4.39420540494523875112274456621, 5.15411590887919422108927548015, 5.78626093991989364522009679404, 6.85144987184122140187168717816, 7.60306988055955565213728264099, 8.033231543054908369325822512442, 9.098676849381107850512645279103, 10.24612130210152021279722077232