L(s) = 1 | − 0.639i·3-s − i·5-s − 7-s + 2.59·9-s − 4.91i·11-s − 5.17i·13-s − 0.639·15-s + 6.70·17-s + 7.89i·19-s + 0.639i·21-s + 1.23·23-s − 25-s − 3.57i·27-s + 2.97i·29-s + 3.83·31-s + ⋯ |
L(s) = 1 | − 0.369i·3-s − 0.447i·5-s − 0.377·7-s + 0.863·9-s − 1.48i·11-s − 1.43i·13-s − 0.165·15-s + 1.62·17-s + 1.81i·19-s + 0.139i·21-s + 0.257·23-s − 0.200·25-s − 0.688i·27-s + 0.553i·29-s + 0.688·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.762679171\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.762679171\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 0.639iT - 3T^{2} \) |
| 11 | \( 1 + 4.91iT - 11T^{2} \) |
| 13 | \( 1 + 5.17iT - 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 - 7.89iT - 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 - 2.97iT - 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 + 5.97iT - 37T^{2} \) |
| 41 | \( 1 + 7.86T + 41T^{2} \) |
| 43 | \( 1 + 0.0848iT - 43T^{2} \) |
| 47 | \( 1 + 9.45T + 47T^{2} \) |
| 53 | \( 1 - 4.91iT - 53T^{2} \) |
| 59 | \( 1 + 0.943iT - 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 + 10.4iT - 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 4.36T + 79T^{2} \) |
| 83 | \( 1 + 2.56iT - 83T^{2} \) |
| 89 | \( 1 - 3.23T + 89T^{2} \) |
| 97 | \( 1 - 4.22T + 97T^{2} \) |
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
−8.579876515332260548940330180809, −8.010815267355792852075155904991, −7.50518812717706274033883963911, −6.27550794839918893083353763636, −5.76206228937961239499331886212, −4.98733527822253179543487154030, −3.55572664389491333133876773090, −3.25160694712255184323815424688, −1.57238513193135068658346931200, −0.66856664259822637024247075622,
1.41496209959524189951845209377, 2.52257259497152096615748414942, 3.58517091563158580788508175758, 4.53731589235166482448295394665, 4.99307355543897504265711238224, 6.38862011275039038644870991918, 7.00029887492266048209982853796, 7.42611053695337364309243449658, 8.609904087548848308437235619519, 9.591501715007160859006944505837