L(s) = 1 | + 2-s − 3-s + 4-s + 1.37·5-s − 6-s − 7-s + 8-s + 9-s + 1.37·10-s + 4·11-s − 12-s + 1.37·13-s − 14-s − 1.37·15-s + 16-s − 4.74·17-s + 18-s + 4·19-s + 1.37·20-s + 21-s + 4·22-s − 23-s − 24-s − 3.11·25-s + 1.37·26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.613·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.433·10-s + 1.20·11-s − 0.288·12-s + 0.380·13-s − 0.267·14-s − 0.354·15-s + 0.250·16-s − 1.15·17-s + 0.235·18-s + 0.917·19-s + 0.306·20-s + 0.218·21-s + 0.852·22-s − 0.208·23-s − 0.204·24-s − 0.623·25-s + 0.269·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.381763565\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.381763565\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 1.37T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 - 6.11T + 43T^{2} \) |
| 47 | \( 1 - 4.62T + 47T^{2} \) |
| 53 | \( 1 - 4.74T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 + 9.37T + 97T^{2} \) |
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
−10.09124647598437864200465143322, −9.366345500486942964371074804205, −8.383255742330188319790166595633, −7.05251170518586249791462921764, −6.41104774704304517006593120815, −5.84065065123578747693183381305, −4.72069752741763663242035470103, −3.91111775644876576270518825142, −2.63902010255096877862274483980, −1.25781928333410151899553950102,
1.25781928333410151899553950102, 2.63902010255096877862274483980, 3.91111775644876576270518825142, 4.72069752741763663242035470103, 5.84065065123578747693183381305, 6.41104774704304517006593120815, 7.05251170518586249791462921764, 8.383255742330188319790166595633, 9.366345500486942964371074804205, 10.09124647598437864200465143322