L(s) = 1 | − 2.56i·2-s − 4.56·4-s + i·7-s + 6.56i·8-s + 1.56·11-s + 0.438i·13-s + 2.56·14-s + 7.68·16-s − 0.438i·17-s + 7.12·19-s − 4i·22-s − 3.12i·23-s + 1.12·26-s − 4.56i·28-s + 6.68·29-s + ⋯ |
L(s) = 1 | − 1.81i·2-s − 2.28·4-s + 0.377i·7-s + 2.31i·8-s + 0.470·11-s + 0.121i·13-s + 0.684·14-s + 1.92·16-s − 0.106i·17-s + 1.63·19-s − 0.852i·22-s − 0.651i·23-s + 0.220·26-s − 0.862i·28-s + 1.24·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.458989014\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.458989014\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 2 | \( 1 + 2.56iT - 2T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.438iT - 13T^{2} \) |
| 17 | \( 1 + 0.438iT - 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 3.12iT - 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 - 0.876iT - 43T^{2} \) |
| 47 | \( 1 + 8.68iT - 47T^{2} \) |
| 53 | \( 1 - 5.12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 10.2iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 2.43T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 + 5.80iT - 97T^{2} \) |
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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
−9.238757062415994590768568222518, −8.780462854672879536799183416853, −7.77637424750418866710603704284, −6.60600736009605813818825208620, −5.40117648155390010991853651392, −4.65637395094068052694216196743, −3.65525108501171721566247332147, −2.90056083926963675153801007478, −1.89732271496415903857570524637, −0.74958019935255402016936242758,
1.06934631818962637403639134847, 3.20212164108774119709264941368, 4.23148591633908601008736655222, 5.08933634328711145828021610982, 5.78480150654248687035572992558, 6.71644333305596354895716346394, 7.21069206004012773186550393530, 8.048228591827209746567656593584, 8.633007312118599913008704802355, 9.614196310615314455188974020870