L(s) = 1 | + (−0.5 + 0.866i)5-s + (1.5 + 2.59i)7-s + (2.5 + 4.33i)11-s + (2.5 − 4.33i)13-s + 2·17-s − 4·19-s + (−0.5 + 0.866i)23-s + (2 + 3.46i)25-s + (−4.5 − 7.79i)29-s + (0.5 − 0.866i)31-s − 3·35-s − 6·37-s + (1.5 − 2.59i)41-s + (−0.5 − 0.866i)43-s + (−1.5 − 2.59i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.566 + 0.981i)7-s + (0.753 + 1.30i)11-s + (0.693 − 1.20i)13-s + 0.485·17-s − 0.917·19-s + (−0.104 + 0.180i)23-s + (0.400 + 0.692i)25-s + (−0.835 − 1.44i)29-s + (0.0898 − 0.155i)31-s − 0.507·35-s − 0.986·37-s + (0.234 − 0.405i)41-s + (−0.0762 − 0.132i)43-s + (−0.218 − 0.378i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18139 + 0.429991i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18139 + 0.429991i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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
−12.33301225841225625353032989756, −11.54674992129962279971637330726, −10.54941577732228680779928163285, −9.482776687426013354812159507706, −8.428057998910143146019568767755, −7.47582765709396604650330576333, −6.21757808552297405855662949323, −5.10502125316642197607072094461, −3.67068864511486347733686858899, −2.03593968508812365837195667138,
1.30484125328406603231345639004, 3.62592093821060769785626162041, 4.55642019763593254059847043674, 6.08959875774201484024840710972, 7.12232559630798562910267941350, 8.421991505255167162476740288671, 8.999481552695027050128102820440, 10.52469225284218391935921330481, 11.19138873605400697127518679357, 12.08951976212825155123198909488