L(s) = 1 | + (0.724 + 1.21i)2-s + (−0.119 + 1.72i)3-s + (−0.950 + 1.75i)4-s + (−0.793 + 0.793i)5-s + (−2.18 + 1.10i)6-s − 7-s + (−2.82 + 0.119i)8-s + (−2.97 − 0.414i)9-s + (−1.53 − 0.389i)10-s + (0.687 + 0.687i)11-s + (−2.92 − 1.85i)12-s + (2.91 − 2.91i)13-s + (−0.724 − 1.21i)14-s + (−1.27 − 1.46i)15-s + (−2.19 − 3.34i)16-s + 1.81i·17-s + ⋯ |
L(s) = 1 | + (0.512 + 0.858i)2-s + (−0.0691 + 0.997i)3-s + (−0.475 + 0.879i)4-s + (−0.354 + 0.354i)5-s + (−0.892 + 0.451i)6-s − 0.377·7-s + (−0.999 + 0.0423i)8-s + (−0.990 − 0.138i)9-s + (−0.486 − 0.123i)10-s + (0.207 + 0.207i)11-s + (−0.844 − 0.535i)12-s + (0.807 − 0.807i)13-s + (−0.193 − 0.324i)14-s + (−0.329 − 0.378i)15-s + (−0.548 − 0.836i)16-s + 0.441i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0739191 - 1.24966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0739191 - 1.24966i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.724 - 1.21i)T \) |
| 3 | \( 1 + (0.119 - 1.72i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (0.793 - 0.793i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.687 - 0.687i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.91 + 2.91i)T - 13iT^{2} \) |
| 17 | \( 1 - 1.81iT - 17T^{2} \) |
| 19 | \( 1 + (-3.24 - 3.24i)T + 19iT^{2} \) |
| 23 | \( 1 - 5.92iT - 23T^{2} \) |
| 29 | \( 1 + (1.54 + 1.54i)T + 29iT^{2} \) |
| 31 | \( 1 - 3.60iT - 31T^{2} \) |
| 37 | \( 1 + (5.10 + 5.10i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.19T + 41T^{2} \) |
| 43 | \( 1 + (7.00 - 7.00i)T - 43iT^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + (-2.08 + 2.08i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.919 - 0.919i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.21 + 4.21i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7.87 - 7.87i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.99iT - 71T^{2} \) |
| 73 | \( 1 - 4.19iT - 73T^{2} \) |
| 79 | \( 1 - 8.07iT - 79T^{2} \) |
| 83 | \( 1 + (-11.0 + 11.0i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.73T + 89T^{2} \) |
| 97 | \( 1 - 6.61T + 97T^{2} \) |
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
−12.02139774131145076069051311540, −11.16329429101377276638398938283, −10.09298377747580734983046736922, −9.161998096594124528627511781147, −8.204529663657355047874911711566, −7.22340570901653917946695914186, −5.96044540885714020629741041163, −5.27608630780174194021117672326, −3.80889398752449458479241277579, −3.32547637125642759844221483816,
0.78198810646755960366245490275, 2.35022458746941016024459843528, 3.66583644424518246540158551161, 4.96677156316074381993108325643, 6.16578612232567527700554089822, 6.99877646653815736643902546413, 8.499791323254509599242596029159, 9.110105295481341477647875427847, 10.43077264324450768257927477806, 11.45696498524193382831495972705