| L(s) = 1 | + (1 + i)3-s + (−1 + i)7-s − i·9-s − 4i·11-s + (−4 + 4i)13-s + (−4 − 4i)17-s − 4·19-s − 2·21-s + (−5 − 5i)23-s + (4 − 4i)27-s + 2i·29-s − 8i·31-s + (4 − 4i)33-s − 8·39-s − 4·41-s + ⋯ |
| L(s) = 1 | + (0.577 + 0.577i)3-s + (−0.377 + 0.377i)7-s − 0.333i·9-s − 1.20i·11-s + (−1.10 + 1.10i)13-s + (−0.970 − 0.970i)17-s − 0.917·19-s − 0.436·21-s + (−1.04 − 1.04i)23-s + (0.769 − 0.769i)27-s + 0.371i·29-s − 1.43i·31-s + (0.696 − 0.696i)33-s − 1.28·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6058311848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6058311848\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + (4 - 4i)T - 13iT^{2} \) |
| 17 | \( 1 + (4 + 4i)T + 17iT^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (5 + 5i)T + 23iT^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + (-7 - 7i)T + 43iT^{2} \) |
| 47 | \( 1 + (3 - 3i)T - 47iT^{2} \) |
| 53 | \( 1 + (4 - 4i)T - 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (3 - 3i)T - 67iT^{2} \) |
| 71 | \( 1 + 16iT - 71T^{2} \) |
| 73 | \( 1 + (-4 + 4i)T - 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (5 + 5i)T + 83iT^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 + (-12 - 12i)T + 97iT^{2} \) |
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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
−9.192900721160815985217300973106, −8.608609749428778522541462133173, −7.67151786192096691761061187536, −6.51233916545006477869457783845, −6.11105529258979221719144960840, −4.67765169022726407765509898463, −4.15087887146774025401614048191, −2.97357296964952183081668291802, −2.27037120910283167330980891325, −0.19487628435779945869852605905,
1.78525176044017764225093375829, 2.46311500184935181434349186261, 3.69070812349028980362855878864, 4.65910509730366936616086594755, 5.57524113552240322005604675061, 6.80379654127223642158764400275, 7.23639904030759459189969826805, 8.089710328410930518911263394814, 8.651729898430027713675274265469, 9.878476042868914690461753608833
