| L(s) = 1 | + (1.57 + 1.59i)5-s + 4.32·7-s + (−0.180 + 0.555i)11-s + (0.298 + 0.918i)13-s + (−1.88 + 1.36i)17-s + (4.35 − 3.16i)19-s + (0.419 − 1.29i)23-s + (−0.0610 + 4.99i)25-s + (−0.571 − 0.415i)29-s + (−6.86 + 4.98i)31-s + (6.79 + 6.87i)35-s + (−1.89 − 5.81i)37-s + (−3.41 − 10.5i)41-s − 7.03·43-s + (7.33 + 5.33i)47-s + ⋯ |
| L(s) = 1 | + (0.702 + 0.711i)5-s + 1.63·7-s + (−0.0544 + 0.167i)11-s + (0.0828 + 0.254i)13-s + (−0.456 + 0.331i)17-s + (0.998 − 0.725i)19-s + (0.0875 − 0.269i)23-s + (−0.0122 + 0.999i)25-s + (−0.106 − 0.0770i)29-s + (−1.23 + 0.896i)31-s + (1.14 + 1.16i)35-s + (−0.310 − 0.956i)37-s + (−0.533 − 1.64i)41-s − 1.07·43-s + (1.07 + 0.777i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.03446 + 0.577625i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03446 + 0.577625i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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
−10.43911116823200755958406121154, −9.227216069383462275126317781289, −8.639038110162420775635828426434, −7.45435044853236298601249170392, −6.97853711304760422396235882246, −5.66364453854198570735620443778, −5.05338232207459461961545067865, −3.88523724435124176850204312914, −2.47580186984116688983883340690, −1.54677608512200492421392741840,
1.21906195067590137485646024724, 2.17239005862360490005288114578, 3.77764405622720344032478312258, 5.11429667261929857863363296919, 5.24948090571229224483518100307, 6.53820690371183460137273147968, 7.79767947587657965071661262570, 8.255652331016441151375261421469, 9.195432913891816586562215364518, 9.953551565995144740666263332746
