L(s) = 1 | + (−0.563 + 0.826i)3-s + (−0.974 − 0.222i)4-s + (−0.930 − 0.365i)7-s + (−0.365 − 0.930i)9-s + (0.733 − 0.680i)12-s + (−0.294 + 0.955i)13-s + (0.900 + 0.433i)16-s + (−0.206 − 0.772i)19-s + (0.826 − 0.563i)21-s + (0.930 − 0.365i)25-s + (0.974 + 0.222i)27-s + (0.826 + 0.563i)28-s + (−0.438 − 1.63i)31-s + (0.149 + 0.988i)36-s + (1.49 + 0.940i)37-s + ⋯ |
L(s) = 1 | + (−0.563 + 0.826i)3-s + (−0.974 − 0.222i)4-s + (−0.930 − 0.365i)7-s + (−0.365 − 0.930i)9-s + (0.733 − 0.680i)12-s + (−0.294 + 0.955i)13-s + (0.900 + 0.433i)16-s + (−0.206 − 0.772i)19-s + (0.826 − 0.563i)21-s + (0.930 − 0.365i)25-s + (0.974 + 0.222i)27-s + (0.826 + 0.563i)28-s + (−0.438 − 1.63i)31-s + (0.149 + 0.988i)36-s + (1.49 + 0.940i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5775468446\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5775468446\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.563 - 0.826i)T \) |
| 7 | \( 1 + (0.930 + 0.365i)T \) |
| 13 | \( 1 + (0.294 - 0.955i)T \) |
good | 2 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 5 | \( 1 + (-0.930 + 0.365i)T^{2} \) |
| 11 | \( 1 + (0.680 - 0.733i)T^{2} \) |
| 17 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 19 | \( 1 + (0.206 + 0.772i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 29 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 31 | \( 1 + (0.438 + 1.63i)T + (-0.866 + 0.5i)T^{2} \) |
| 37 | \( 1 + (-1.49 - 0.940i)T + (0.433 + 0.900i)T^{2} \) |
| 41 | \( 1 + (0.149 - 0.988i)T^{2} \) |
| 43 | \( 1 + (-1.85 + 0.139i)T + (0.988 - 0.149i)T^{2} \) |
| 47 | \( 1 + (-0.680 + 0.733i)T^{2} \) |
| 53 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 59 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 61 | \( 1 + (-0.432 + 1.40i)T + (-0.826 - 0.563i)T^{2} \) |
| 67 | \( 1 + (-0.488 + 1.82i)T + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.563 - 0.826i)T^{2} \) |
| 73 | \( 1 + (0.799 - 1.83i)T + (-0.680 - 0.733i)T^{2} \) |
| 79 | \( 1 + (-0.974 - 1.68i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 89 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 97 | \( 1 + (-1.77 - 0.474i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.432516117578416153890256800006, −9.041668459817913007639074913878, −7.939808154617566105532523218142, −6.76459255122640045900276777931, −6.16778747928853324681873921742, −5.23565027978708710594763965252, −4.36317586304844819470952779904, −3.94166112762753124665988402406, −2.72699782388362268679401445283, −0.66200065159909362692704804930,
0.916018502266066632276688399408, 2.57459050455558449000182842289, 3.48031050487775218882478980031, 4.65811131764257318747413394117, 5.57324466198587284941521911202, 6.04156938293624426987808665814, 7.16020614947644678192126739408, 7.75262841182234544421579492385, 8.667134484569083415372921101404, 9.243193346398021406613723627543