L(s) = 1 | + (−0.888 − 0.458i)3-s + (0.981 − 0.189i)4-s + (0.415 + 0.909i)7-s + (0.580 + 0.814i)9-s + (−0.959 − 0.281i)12-s + (0.601 − 0.573i)13-s + (0.928 − 0.371i)16-s + (−1.03 + 1.44i)19-s + (0.0475 − 0.998i)21-s + (0.235 + 0.971i)25-s + (−0.142 − 0.989i)27-s + (0.580 + 0.814i)28-s + (−0.0671 + 0.276i)31-s + (0.723 + 0.690i)36-s + (−0.235 − 0.408i)37-s + ⋯ |
L(s) = 1 | + (−0.888 − 0.458i)3-s + (0.981 − 0.189i)4-s + (0.415 + 0.909i)7-s + (0.580 + 0.814i)9-s + (−0.959 − 0.281i)12-s + (0.601 − 0.573i)13-s + (0.928 − 0.371i)16-s + (−1.03 + 1.44i)19-s + (0.0475 − 0.998i)21-s + (0.235 + 0.971i)25-s + (−0.142 − 0.989i)27-s + (0.580 + 0.814i)28-s + (−0.0671 + 0.276i)31-s + (0.723 + 0.690i)36-s + (−0.235 − 0.408i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.106975021\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.106975021\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.888 + 0.458i)T \) |
| 7 | \( 1 + (-0.415 - 0.909i)T \) |
| 67 | \( 1 + (0.959 + 0.281i)T \) |
good | 2 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 5 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 11 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 13 | \( 1 + (-0.601 + 0.573i)T + (0.0475 - 0.998i)T^{2} \) |
| 17 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (1.03 - 1.44i)T + (-0.327 - 0.945i)T^{2} \) |
| 23 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.0671 - 0.276i)T + (-0.888 - 0.458i)T^{2} \) |
| 37 | \( 1 + (0.235 + 0.408i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 43 | \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 53 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 59 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 61 | \( 1 + (-1.39 + 1.09i)T + (0.235 - 0.971i)T^{2} \) |
| 71 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 73 | \( 1 + (0.264 + 1.83i)T + (-0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (0.452 + 0.132i)T + (0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 89 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 97 | \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \) |
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
−10.05194207212809251720007234247, −8.812083850537143392617203610005, −7.971869843473107868846921018508, −7.27347739297084454004147722739, −6.26063939756579625155331219956, −5.81127286444563675266329956228, −5.09595693132725506974963557356, −3.66461522392361870463652220119, −2.30098059330526078840131179164, −1.47929762579004499911982925693,
1.20099174954631601048458790066, 2.62925145468953252965656926339, 3.99658216325276118461378987824, 4.54285290839912422321502919718, 5.73775641852957166136732546475, 6.64146248321226310750212304194, 6.97750872042254562110571262422, 8.053311510752162561150180772870, 8.987875555738077227507700346557, 10.07488098590250541679003054715