L(s) = 1 | + (−0.707 + 0.707i)2-s − i·3-s − 1.00i·4-s + (−0.0951 − 0.0951i)5-s + (0.707 + 0.707i)6-s + (−0.0951 − 2.64i)7-s + (0.707 + 0.707i)8-s − 9-s + 0.134·10-s + (−3.64 − 3.64i)11-s − 1.00·12-s + (−2 + 3i)13-s + (1.93 + 1.80i)14-s + (−0.0951 + 0.0951i)15-s − 1.00·16-s − 5.98·17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.577i·3-s − 0.500i·4-s + (−0.0425 − 0.0425i)5-s + (0.288 + 0.288i)6-s + (−0.0359 − 0.999i)7-s + (0.250 + 0.250i)8-s − 0.333·9-s + 0.0425·10-s + (−1.09 − 1.09i)11-s − 0.288·12-s + (−0.554 + 0.832i)13-s + (0.517 + 0.481i)14-s + (−0.0245 + 0.0245i)15-s − 0.250·16-s − 1.45·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.106164 - 0.393231i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.106164 - 0.393231i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (0.0951 + 2.64i)T \) |
| 13 | \( 1 + (2 - 3i)T \) |
good | 5 | \( 1 + (0.0951 + 0.0951i)T + 5iT^{2} \) |
| 11 | \( 1 + (3.64 + 3.64i)T + 11iT^{2} \) |
| 17 | \( 1 + 5.98T + 17T^{2} \) |
| 19 | \( 1 + (-4.19 - 4.19i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.69iT - 23T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 + (-0.739 - 0.739i)T + 31iT^{2} \) |
| 37 | \( 1 + (4.83 + 4.83i)T + 37iT^{2} \) |
| 41 | \( 1 + (3.04 + 3.04i)T + 41iT^{2} \) |
| 43 | \( 1 + 8.78iT - 43T^{2} \) |
| 47 | \( 1 + (3.28 - 3.28i)T - 47iT^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 + (1.30 - 1.30i)T - 59iT^{2} \) |
| 61 | \( 1 + 5.98iT - 61T^{2} \) |
| 67 | \( 1 + (0.0454 - 0.0454i)T - 67iT^{2} \) |
| 71 | \( 1 + (-8.69 + 8.69i)T - 71iT^{2} \) |
| 73 | \( 1 + (-4.83 + 4.83i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + (-1.28 - 1.28i)T + 83iT^{2} \) |
| 89 | \( 1 + (-9.21 + 9.21i)T - 89iT^{2} \) |
| 97 | \( 1 + (-9.97 - 9.97i)T + 97iT^{2} \) |
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
−10.46341929613216769902638781000, −9.445597193658361028293290128287, −8.489302041439476499159027441428, −7.64788298255783038916104359940, −7.02931077715164528421084172991, −6.02164165475908481713071787597, −4.99013545398374012889155285276, −3.59749226875040220013926904144, −1.96402794558974565065255035754, −0.25736904524656647619023009571,
2.26252672335382339710729988371, 3.04164149823348289147813919029, 4.68593743647949807625755054106, 5.30043314097502105756708201574, 6.80402720937937783962777631325, 7.80749448790132785244048560435, 8.733692675625309829516145923405, 9.528152986973598342440522372011, 10.17652912258863932722572432350, 11.10233095268026542429704494072