L(s) = 1 | + (−1 + i)3-s + (1 + i)7-s + i·9-s − 6i·11-s + (−1 − i)13-s + (−1 + i)17-s − 4·19-s − 2·21-s + (−5 + 5i)23-s + (−4 − 4i)27-s + 8i·29-s − 2i·31-s + (6 + 6i)33-s + (−5 + 5i)37-s + 2·39-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.577i)3-s + (0.377 + 0.377i)7-s + 0.333i·9-s − 1.80i·11-s + (−0.277 − 0.277i)13-s + (−0.242 + 0.242i)17-s − 0.917·19-s − 0.436·21-s + (−1.04 + 1.04i)23-s + (−0.769 − 0.769i)27-s + 1.48i·29-s − 0.359i·31-s + (1.04 + 1.04i)33-s + (−0.821 + 0.821i)37-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1 - i)T + 7iT^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (1 - i)T - 17iT^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (5 - 5i)T - 23iT^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (-3 + 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (7 + 7i)T + 47iT^{2} \) |
| 53 | \( 1 + (1 + i)T + 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (7 + 7i)T + 67iT^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + (9 + 9i)T + 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (5 - 5i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
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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
−8.948862844657013768046769506814, −8.406837679272210433957835242484, −7.63282397342565243643138374584, −6.39077459417847177487225285081, −5.67345537595415015490197509243, −5.11131520097096109500924912143, −4.05225121294251739274196101591, −3.10534604220966593671916779248, −1.78081875858630517063994731620, 0,
1.56730681119818120057759854084, 2.48054409441270760871203712167, 4.24887415762888992609219846950, 4.51785488697542070280998210141, 5.86575297264700713333873134176, 6.55940039953861065185824892891, 7.28854716051763515581253346644, 7.889536571156655130389814405102, 9.025637204908989211419191621093