L(s) = 1 | + 3.98i·2-s − 3i·3-s − 7.89·4-s + 11.9·6-s − 12.5i·7-s + 0.411i·8-s − 9·9-s − 11·11-s + 23.6i·12-s + 36.0i·13-s + 50.0·14-s − 64.8·16-s − 39.7i·17-s − 35.8i·18-s + 148.·19-s + ⋯ |
L(s) = 1 | + 1.40i·2-s − 0.577i·3-s − 0.987·4-s + 0.813·6-s − 0.678i·7-s + 0.0181i·8-s − 0.333·9-s − 0.301·11-s + 0.569i·12-s + 0.769i·13-s + 0.956·14-s − 1.01·16-s − 0.566i·17-s − 0.469i·18-s + 1.78·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.886820050\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.886820050\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 - 3.98iT - 8T^{2} \) |
| 7 | \( 1 + 12.5iT - 343T^{2} \) |
| 13 | \( 1 - 36.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 39.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 35.0iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 88.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 85.2iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 329.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 278. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 272. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 223. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 467.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 752.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 733. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 537.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 397. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 683. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 166.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 694. iT - 9.12e5T^{2} \) |
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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
−9.603250651142975161983090216394, −8.954805418902562309625107368875, −7.83234414258475663451590834541, −7.35184909413400720651458229847, −6.77803104210051312211384405987, −5.72704114743985802645264387260, −5.02752633210828710084753412813, −3.79764942528097243540096891081, −2.29742819765406902197052617667, −0.73512580128861731747000591073,
0.792406293486479651168939452229, 2.13763025337107342144291924396, 3.11903028185857041871829819008, 3.81542662959316823262937669299, 5.09886496719466505690602612810, 5.78552982553888537594691408165, 7.24273256107432825296812635116, 8.300741559875148001821090908743, 9.282749176372629978862726625527, 9.807993966665738649969142595518