L(s) = 1 | + (−0.707 + 0.707i)2-s + 0.999i·4-s + (−0.765 + 1.84i)5-s + (1.53 + 3.69i)7-s + (−2.12 − 2.12i)8-s + (−2.12 − 2.12i)9-s + (−0.765 − 1.84i)10-s − 2i·13-s + (−3.69 − 1.53i)14-s + 1.00·16-s + 3·18-s + (−2.82 + 2.82i)19-s + (−1.84 − 0.765i)20-s + (−3.69 + 1.53i)23-s + (0.707 + 0.707i)25-s + (1.41 + 1.41i)26-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + 0.499i·4-s + (−0.342 + 0.826i)5-s + (0.578 + 1.39i)7-s + (−0.750 − 0.750i)8-s + (−0.707 − 0.707i)9-s + (−0.242 − 0.584i)10-s − 0.554i·13-s + (−0.987 − 0.409i)14-s + 0.250·16-s + 0.707·18-s + (−0.648 + 0.648i)19-s + (−0.413 − 0.171i)20-s + (−0.770 + 0.319i)23-s + (0.141 + 0.141i)25-s + (0.277 + 0.277i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.100374 + 0.713784i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.100374 + 0.713784i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 + (0.707 - 0.707i)T - 2iT^{2} \) |
| 3 | \( 1 + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (0.765 - 1.84i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1.53 - 3.69i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 + (2.82 - 2.82i)T - 19iT^{2} \) |
| 23 | \( 1 + (3.69 - 1.53i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (2.29 - 5.54i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-3.69 - 1.53i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-1.84 - 0.765i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (2.29 + 5.54i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-2.82 - 2.82i)T + 43iT^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \) |
| 59 | \( 1 + (-8.48 - 8.48i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.82 - 9.23i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + (-3.69 - 1.53i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (2.29 - 5.54i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-11.0 + 4.59i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.82 + 2.82i)T - 83iT^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 + (0.765 - 1.84i)T + (-68.5 - 68.5i)T^{2} \) |
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
−12.01920615652872121804871393327, −11.52903958196578572842170272916, −10.28143263741546218469409409540, −8.981760305622564811521478854140, −8.483066282495300592345985042133, −7.53628591221710317050915731345, −6.42371734545121169417803174997, −5.54463440645621299841802258031, −3.66768610800993715881367741692, −2.62910998232784470428584633312,
0.62497419398261959914472369781, 2.16933983264778584539643925041, 4.21639436182380443339416007427, 5.04703290419720831473774142565, 6.42635013392396773270046341939, 7.87065804890524287231228230036, 8.505355987977238323301935903270, 9.593285546609606616389732600868, 10.57055621693463971554917827170, 11.19404335646009565083133915810