L(s) = 1 | + (−1.22 + 0.707i)2-s + (−1 + 1.41i)3-s + (0.999 − 1.73i)4-s + 1.41i·5-s + (0.224 − 2.43i)6-s + (−2 + 1.73i)7-s + 2.82i·8-s + (−1.00 − 2.82i)9-s + (−1.00 − 1.73i)10-s − 2.44·11-s + (1.44 + 3.14i)12-s + 2·13-s + (1.22 − 3.53i)14-s + (−2.00 − 1.41i)15-s + (−2.00 − 3.46i)16-s − 7.34·17-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)2-s + (−0.577 + 0.816i)3-s + (0.499 − 0.866i)4-s + 0.632i·5-s + (0.0917 − 0.995i)6-s + (−0.755 + 0.654i)7-s + 0.999i·8-s + (−0.333 − 0.942i)9-s + (−0.316 − 0.547i)10-s − 0.738·11-s + (0.418 + 0.908i)12-s + 0.554·13-s + (0.327 − 0.944i)14-s + (−0.516 − 0.365i)15-s + (−0.500 − 0.866i)16-s − 1.78·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0175692 - 0.357375i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0175692 - 0.357375i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 7.34T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 - 6.92iT - 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 5.65iT - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 1.41iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 11.3iT - 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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
−13.35631869730602476246417718451, −11.97689333775247697843858883303, −10.80948629571514510457949666458, −10.45049429757349253245543752179, −9.225576258704113753876125665914, −8.496454584853849515614411450914, −6.72257724063785497298241506341, −6.19893415291904211512770343297, −4.82154067556283219880331755631, −2.82069919849367578603505346585,
0.44602461352603344099013286196, 2.34498505449897730852703937755, 4.32968932380572203867346331402, 6.20499814780678485213621427714, 7.10673328379375224030051821566, 8.242665093089288828162320377154, 9.146748128474840829353380223943, 10.60310383860426572522012689273, 11.01776639433292529771665194211, 12.39695898305542850184630905677