L(s) = 1 | − 8i·7-s − 36·11-s − 10i·13-s + 18i·17-s + 100·19-s − 72i·23-s − 234·29-s − 16·31-s + 226i·37-s − 90·41-s + 452i·43-s + 432i·47-s + 279·49-s − 414i·53-s − 684·59-s + ⋯ |
L(s) = 1 | − 0.431i·7-s − 0.986·11-s − 0.213i·13-s + 0.256i·17-s + 1.20·19-s − 0.652i·23-s − 1.49·29-s − 0.0926·31-s + 1.00i·37-s − 0.342·41-s + 1.60i·43-s + 1.34i·47-s + 0.813·49-s − 1.07i·53-s − 1.50·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7720684998\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7720684998\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 8iT - 343T^{2} \) |
| 11 | \( 1 + 36T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 18iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 100T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 234T + 2.43e4T^{2} \) |
| 31 | \( 1 + 16T + 2.97e4T^{2} \) |
| 37 | \( 1 - 226iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 90T + 6.89e4T^{2} \) |
| 43 | \( 1 - 452iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 432iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 414iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 684T + 2.05e5T^{2} \) |
| 61 | \( 1 - 422T + 2.26e5T^{2} \) |
| 67 | \( 1 + 332iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 360T + 3.57e5T^{2} \) |
| 73 | \( 1 - 26iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 512T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 630T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3iT - 9.12e5T^{2} \) |
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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
−9.987229847604555702037506687451, −9.289807869095835878138461353066, −8.099282683035357314673498208106, −7.62361154736511932299110106330, −6.60130456414650682455309609079, −5.56943809869186732648341564406, −4.76575233532445731382951806425, −3.59557748071621097046319719558, −2.59837407138362409218810670935, −1.18012993954132801611757528065,
0.20640262241123807863596677616, 1.80137887952025724190745892935, 2.90509752027566809105551137149, 3.95981512948388568702326972753, 5.32965574008324861582784947859, 5.65129207613844891936159037045, 7.11960714184046959892873589543, 7.61779307590673668924201597645, 8.712223993140354364054080434338, 9.415862468905478396139051004638