L(s) = 1 | + (−0.755 + 0.654i)2-s + (0.989 + 0.142i)3-s + (0.142 − 0.989i)4-s + (−0.281 − 0.959i)5-s + (−0.841 + 0.540i)6-s + (0.540 + 0.841i)8-s + (0.959 + 0.281i)9-s + (0.841 + 0.540i)10-s + (0.281 − 0.959i)12-s + (−0.142 − 0.989i)15-s + (−0.959 − 0.281i)16-s + (−0.755 − 1.65i)17-s + (−0.909 + 0.415i)18-s + (1.37 + 0.627i)19-s + (−0.989 + 0.142i)20-s + ⋯ |
L(s) = 1 | + (−0.755 + 0.654i)2-s + (0.989 + 0.142i)3-s + (0.142 − 0.989i)4-s + (−0.281 − 0.959i)5-s + (−0.841 + 0.540i)6-s + (0.540 + 0.841i)8-s + (0.959 + 0.281i)9-s + (0.841 + 0.540i)10-s + (0.281 − 0.959i)12-s + (−0.142 − 0.989i)15-s + (−0.959 − 0.281i)16-s + (−0.755 − 1.65i)17-s + (−0.909 + 0.415i)18-s + (1.37 + 0.627i)19-s + (−0.989 + 0.142i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.155706968\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.155706968\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.755 - 0.654i)T \) |
| 3 | \( 1 + (-0.989 - 0.142i)T \) |
| 5 | \( 1 + (0.281 + 0.959i)T \) |
| 23 | \( 1 + (-0.540 + 0.841i)T \) |
good | 7 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 11 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (0.755 + 1.65i)T + (-0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (-1.37 - 0.627i)T + (0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 43 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + 1.08T + T^{2} \) |
| 53 | \( 1 + (1.27 - 1.10i)T + (0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 61 | \( 1 + (-1.07 + 0.153i)T + (0.959 - 0.281i)T^{2} \) |
| 67 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 79 | \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 83 | \( 1 + (0.0801 - 0.273i)T + (-0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
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
−8.986943058363802941161956833161, −8.208675275851849469124039439524, −7.62081828226324521234775719467, −7.07041736616103721511810031171, −5.97773272865695977876592017297, −4.93110711103778093483324029324, −4.52235747648954249867626274703, −3.20175781160177735125275088814, −2.10609044586986062374687933185, −0.922270917603345922997764293100,
1.47685889694227359458338880898, 2.37094238805826448339413195070, 3.40030934486086521218194194514, 3.63060532301289903058706385289, 4.89437453834479952920020195466, 6.46511652290204364377401447291, 7.00687553717746435096772394961, 7.70596217359178142346864369339, 8.360544523443787628093849018986, 9.001374603529011484419282713321