L(s) = 1 | − i·2-s − 3-s − 4-s − i·5-s + i·6-s + 3.37·7-s + i·8-s + 9-s − 10-s + 0.627·11-s + 12-s + 6.74i·13-s − 3.37i·14-s + i·15-s + 16-s − 5.37i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s + 1.27·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.189·11-s + 0.288·12-s + 1.87i·13-s − 0.901i·14-s + 0.258i·15-s + 0.250·16-s − 1.30i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.428608840\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.428608840\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 37 | \( 1 + (-5.74 - 2i)T \) |
good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 0.627T + 11T^{2} \) |
| 13 | \( 1 - 6.74iT - 13T^{2} \) |
| 17 | \( 1 + 5.37iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 2.74iT - 23T^{2} \) |
| 29 | \( 1 - 9.37iT - 29T^{2} \) |
| 31 | \( 1 + 2.62iT - 31T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 - 1.37iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.62T + 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 11.3iT - 61T^{2} \) |
| 67 | \( 1 + 2.74T + 67T^{2} \) |
| 71 | \( 1 - 2.74T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 4.74iT - 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 8.74iT - 89T^{2} \) |
| 97 | \( 1 - 10.1iT - 97T^{2} \) |
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
−9.649695161398348080969765072009, −9.283104299790638226268508871713, −8.231895855096532150250729462593, −7.42474552646136919839328943487, −6.34508692076427384538356634429, −5.16633140900496099770904950348, −4.66182876990648648659780735521, −3.74108494948740091375639066757, −2.04617043186955052968529706958, −1.23524046644936675278577660511,
0.825500563024025079723785534084, 2.52451980465773798333625813292, 4.01936165167348352082623722085, 4.87576498795284858918764425792, 5.73932101764878007164976292127, 6.38061862619589226396654967871, 7.57134068737927469213961562667, 7.969484309908610500951182057075, 8.846289189102558221678703661595, 10.01862324483806408967756458075