L(s) = 1 | + 2-s + i·3-s + 4-s + i·6-s + 5.12·7-s + 8-s − 9-s + 3.12i·11-s + i·12-s + (3.56 − 0.561i)13-s + 5.12·14-s + 16-s − 2i·17-s − 18-s − 6i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 1.93·7-s + 0.353·8-s − 0.333·9-s + 0.941i·11-s + 0.288i·12-s + (0.987 − 0.155i)13-s + 1.36·14-s + 0.250·16-s − 0.485i·17-s − 0.235·18-s − 1.37i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.573644048\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.573644048\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.56 + 0.561i)T \) |
good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 3.12iT - 11T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 3.12iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 5.12iT - 31T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 + 9.12iT - 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 11.3iT - 53T^{2} \) |
| 59 | \( 1 - 7.12iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 6.24iT - 71T^{2} \) |
| 73 | \( 1 + 4.87T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 5.12iT - 89T^{2} \) |
| 97 | \( 1 - 4.87T + 97T^{2} \) |
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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
−9.046532923520976304624704439827, −8.608910184356256243031300992436, −7.52561886941854395807025956389, −7.04145283102416146997108320783, −5.69920461127979733399294528878, −5.05152443058114965860914500734, −4.52733588851498654606123115402, −3.65210333893476663583269135057, −2.39414728000065705917862966864, −1.40705391671701071882904796551,
1.28432688322661936306282395102, 1.94493432342081238329484843558, 3.31142390486135480103896126337, 4.22015431796780892732605626769, 5.09941659616532538390615043899, 5.93324231701598114204043984664, 6.48127925272364125998217491812, 7.81700528020548538854474676703, 8.092580995260058328677590795611, 8.702855377378708212296254382845