| L(s) = 1 | + (0.809 + 0.587i)3-s + (−1.57 − 1.59i)5-s + 4.32·7-s + (0.309 + 0.951i)9-s + (0.180 − 0.555i)11-s + (0.298 + 0.918i)13-s + (−0.336 − 2.21i)15-s + (1.88 − 1.36i)17-s + (4.35 − 3.16i)19-s + (3.49 + 2.54i)21-s + (−0.419 + 1.29i)23-s + (−0.0610 + 4.99i)25-s + (−0.309 + 0.951i)27-s + (0.571 + 0.415i)29-s + (−6.86 + 4.98i)31-s + ⋯ |
| L(s) = 1 | + (0.467 + 0.339i)3-s + (−0.702 − 0.711i)5-s + 1.63·7-s + (0.103 + 0.317i)9-s + (0.0544 − 0.167i)11-s + (0.0828 + 0.254i)13-s + (−0.0868 − 0.570i)15-s + (0.456 − 0.331i)17-s + (0.998 − 0.725i)19-s + (0.763 + 0.554i)21-s + (−0.0875 + 0.269i)23-s + (−0.0122 + 0.999i)25-s + (−0.0594 + 0.183i)27-s + (0.106 + 0.0770i)29-s + (−1.23 + 0.896i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56922 - 0.0589039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56922 - 0.0589039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (1.57 + 1.59i)T \) |
| good | 7 | \( 1 - 4.32T + 7T^{2} \) |
| 11 | \( 1 + (-0.180 + 0.555i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.298 - 0.918i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.88 + 1.36i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.35 + 3.16i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.419 - 1.29i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.571 - 0.415i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (6.86 - 4.98i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.89 + 5.81i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.41 - 10.5i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 7.03T + 43T^{2} \) |
| 47 | \( 1 + (7.33 + 5.33i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (7.20 + 5.23i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.25 + 6.93i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.48 - 4.57i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-0.304 + 0.221i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (8.54 + 6.20i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.0659 + 0.202i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.68 - 4.12i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (13.0 - 9.46i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.33 - 13.3i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (12.7 + 9.25i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.48400174432384306687164945446, −11.13922857034619178826192780500, −9.678678085949047345145319905223, −8.746950512373144541859214616902, −8.033141890037615530822151529868, −7.24370626129737133750959258532, −5.27477633206128880551917841418, −4.68992610641936173285053949771, −3.41697897100048037267751350460, −1.52068892563759896701500530207,
1.69663220115879902833391500232, 3.26521246908136153526298154657, 4.46789721949115305135216121691, 5.79543176415519640722057051460, 7.29001068255349012286176787930, 7.82649401386720081233719966250, 8.593928600619606192985898591592, 9.987397645352122685232857752444, 10.98570956030212669391978243156, 11.69269502291455007787892374364
