L(s) = 1 | + (1.36 − 0.366i)3-s + (0.866 − 0.5i)9-s + 11-s + (−0.366 + 1.36i)13-s + (−0.366 − 1.36i)17-s + (0.866 − 0.5i)19-s + (−0.866 + 0.5i)29-s − 31-s + (1.36 − 0.366i)33-s + 2i·39-s + (1.36 + 0.366i)47-s + i·49-s + (−1 − 1.73i)51-s + (0.366 − 1.36i)53-s + (0.999 − i)57-s + ⋯ |
L(s) = 1 | + (1.36 − 0.366i)3-s + (0.866 − 0.5i)9-s + 11-s + (−0.366 + 1.36i)13-s + (−0.366 − 1.36i)17-s + (0.866 − 0.5i)19-s + (−0.866 + 0.5i)29-s − 31-s + (1.36 − 0.366i)33-s + 2i·39-s + (1.36 + 0.366i)47-s + i·49-s + (−1 − 1.73i)51-s + (0.366 − 1.36i)53-s + (0.999 − i)57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.841652886\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.841652886\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
good | 3 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
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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
−9.186000993643986013578594842835, −8.919121741817829198595992592193, −7.65421144829346351958153048841, −7.22408574963719423001656699894, −6.52546200645533327677722168792, −5.21908947805727914539195924232, −4.23941717552388011520151278704, −3.38574206943600046849459141704, −2.45231505569630955189153346376, −1.53806903695440170300265687993,
1.59155617050334480192927833145, 2.70918433544033429040583093569, 3.62652131941826562328968734346, 4.11426053011071020663282230444, 5.44891611792543243254567145108, 6.20571139395083512838391271242, 7.54287177936144695084707676346, 7.79664069975286063727612546088, 8.964648367930517526376842851956, 9.062530216394892608241402247292