L(s) = 1 | + (1.34 + 0.422i)2-s + 3-s + (1.64 + 1.14i)4-s − 1.97·5-s + (1.34 + 0.422i)6-s + 2.69·7-s + (1.73 + 2.23i)8-s + 9-s + (−2.66 − 0.834i)10-s + 0.719i·11-s + (1.64 + 1.14i)12-s + 0.585i·13-s + (3.63 + 1.13i)14-s − 1.97·15-s + (1.39 + 3.74i)16-s − 5.15i·17-s + ⋯ |
L(s) = 1 | + (0.954 + 0.299i)2-s + 0.577·3-s + (0.821 + 0.570i)4-s − 0.882·5-s + (0.550 + 0.172i)6-s + 1.01·7-s + (0.612 + 0.790i)8-s + 0.333·9-s + (−0.842 − 0.263i)10-s + 0.217i·11-s + (0.474 + 0.329i)12-s + 0.162i·13-s + (0.971 + 0.304i)14-s − 0.509·15-s + (0.348 + 0.937i)16-s − 1.24i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.79066 + 1.01460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.79066 + 1.01460i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.34 - 0.422i)T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + (0.180 + 4.79i)T \) |
good | 5 | \( 1 + 1.97T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 0.719iT - 11T^{2} \) |
| 13 | \( 1 - 0.585iT - 13T^{2} \) |
| 17 | \( 1 + 5.15iT - 17T^{2} \) |
| 19 | \( 1 - 6.29iT - 19T^{2} \) |
| 29 | \( 1 + 1.33iT - 29T^{2} \) |
| 31 | \( 1 - 0.359iT - 31T^{2} \) |
| 37 | \( 1 - 3.15T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 0.124iT - 43T^{2} \) |
| 47 | \( 1 + 7.12iT - 47T^{2} \) |
| 53 | \( 1 - 2.15T + 53T^{2} \) |
| 59 | \( 1 + 7.69T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 13.3iT - 67T^{2} \) |
| 71 | \( 1 + 8.35iT - 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 + 9.57iT - 83T^{2} \) |
| 89 | \( 1 - 13.9iT - 89T^{2} \) |
| 97 | \( 1 + 7.32iT - 97T^{2} \) |
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
−11.18584532390881963251905430747, −10.17004018971933077013809744724, −8.789353161035148036533601100527, −7.86439551468581515278940149931, −7.52765372213540761805625110311, −6.31400412013110048890552631354, −4.99507991913166720784847077189, −4.29140513093107588062058871035, −3.28175282749425885413208225290, −1.94984305073569077266412434125,
1.55003231807090067679984658360, 2.94817426964597972385727330013, 4.00202019574133831089545754450, 4.75230631675683678247914291028, 5.90761307610108744084966520083, 7.18935867346856248846174156722, 7.87801276032988914852450472185, 8.783211453674921623865088795294, 10.02261032942734960263439506483, 11.12583605252016098753954236020