L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (1 − 1.73i)5-s + 0.999·6-s + (−0.5 − 2.59i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.999 − 1.73i)10-s + (−1.5 − 2.59i)11-s + (0.499 − 0.866i)12-s − 13-s + (−2.5 − 0.866i)14-s + 1.99·15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.447 − 0.774i)5-s + 0.408·6-s + (−0.188 − 0.981i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + (−0.452 − 0.783i)11-s + (0.144 − 0.249i)12-s − 0.277·13-s + (−0.668 − 0.231i)14-s + 0.516·15-s + (−0.125 + 0.216i)16-s + (−0.121 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07031 - 1.40690i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07031 - 1.40690i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.5 + 9.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 + (-6 - 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.58948785133078455353567380320, −9.731708604252730298311664446684, −9.039277001127544061606992002705, −8.084624562723368102394740189153, −6.85657401507790752482144449679, −5.52874986029181995125765103422, −4.78112701832998446623982469398, −3.75994407612231324242106755751, −2.63931602917772875355829287061, −0.911323824910058518842134785462,
2.20043645908680964663657032609, 3.05306367969322563837806220540, 4.60167290071852992059786229672, 5.81242054577302468369073997872, 6.41941987325533303280365592014, 7.41041319569221088915125316641, 8.160893328886625656840873248713, 9.253347824562948930149468558988, 9.992170014811414547373982348184, 11.11627975423796464173872017574