L(s) = 1 | − 2.73i·3-s − i·5-s − 0.732·7-s − 4.46·9-s + 2i·11-s − 3.46i·13-s − 2.73·15-s + 3.46·17-s − 0.535i·19-s + 2i·21-s + 6.19·23-s − 25-s + 3.99i·27-s + 6.92i·29-s + 5.46·31-s + ⋯ |
L(s) = 1 | − 1.57i·3-s − 0.447i·5-s − 0.276·7-s − 1.48·9-s + 0.603i·11-s − 0.960i·13-s − 0.705·15-s + 0.840·17-s − 0.122i·19-s + 0.436i·21-s + 1.29·23-s − 0.200·25-s + 0.769i·27-s + 1.28i·29-s + 0.981·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.658367 - 0.858000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.658367 - 0.858000i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 3 | \( 1 + 2.73iT - 3T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 0.535iT - 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 - 5.26iT - 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 - 7.46iT - 59T^{2} \) |
| 61 | \( 1 + 8.92iT - 61T^{2} \) |
| 67 | \( 1 + 10.7iT - 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 1.07T + 79T^{2} \) |
| 83 | \( 1 + 1.26iT - 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−12.66957005593011508255180528453, −12.04759357181650432074593686855, −10.73654620022073340098703779697, −9.387370989585894219543836341344, −8.163922623914325870223683387448, −7.37910142044265702757719372060, −6.34632751921245849871945507548, −5.09947870171537809908534329092, −2.94655939220162838838456849961, −1.19825709651491645034593071036,
3.06159837411850381180474499274, 4.15431509374419544021751980963, 5.38663634296394194511571409250, 6.66801448540019197078717706831, 8.288051536299278320481287814306, 9.410217297651719987564535040589, 10.06797603690172882670540691445, 11.05570888154484132433956799558, 11.81418333633619723700614308237, 13.39013680020442365032820602602