L(s) = 1 | + (1.41 + 1.41i)3-s + (1.41 − 1.41i)5-s − 4i·7-s + 1.00i·9-s + (−1.41 + 1.41i)11-s + (1.41 + 1.41i)13-s + 4.00·15-s + 2·17-s + (−1.41 − 1.41i)19-s + (5.65 − 5.65i)21-s − 4i·23-s + 0.999i·25-s + (2.82 − 2.82i)27-s + (4.24 + 4.24i)29-s − 4.00·33-s + ⋯ |
L(s) = 1 | + (0.816 + 0.816i)3-s + (0.632 − 0.632i)5-s − 1.51i·7-s + 0.333i·9-s + (−0.426 + 0.426i)11-s + (0.392 + 0.392i)13-s + 1.03·15-s + 0.485·17-s + (−0.324 − 0.324i)19-s + (1.23 − 1.23i)21-s − 0.834i·23-s + 0.199i·25-s + (0.544 − 0.544i)27-s + (0.787 + 0.787i)29-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.345013099\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.345013099\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-1.41 - 1.41i)T + 3iT^{2} \) |
| 5 | \( 1 + (-1.41 + 1.41i)T - 5iT^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + (1.41 - 1.41i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.41 - 1.41i)T + 13iT^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + (1.41 + 1.41i)T + 19iT^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + (-4.24 - 4.24i)T + 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-7.07 + 7.07i)T - 37iT^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + (-4.24 + 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \) |
| 59 | \( 1 + (9.89 - 9.89i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.41 + 1.41i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-4.24 - 4.24i)T + 83iT^{2} \) |
| 89 | \( 1 - 2iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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
−9.936203449858407066101766210083, −9.105151362839531963921654237869, −8.474509505206986133625904796930, −7.45948193348215694965451488623, −6.62333895088324518299342854291, −5.35633980321687366441404325791, −4.37316585694665343858710972664, −3.83788875370598494527992325940, −2.56983300073975949157135885651, −1.06757726028612270039533786595,
1.63751757047662800139132892455, 2.65285222605664471768876986082, 3.11869155992713791672063183052, 4.94674293624403602174610833580, 6.04929189231026863666023906150, 6.37714583895726910445856578997, 7.919510699779543160589495179527, 8.056407383876123896830699747140, 9.098806193804102531797053295252, 9.831823326973790446938575509174