| L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−4 + 6.92i)5-s + 7.99·8-s + (−7.99 − 13.8i)10-s + (20 + 34.6i)11-s − 4·13-s + (−8 + 13.8i)16-s + (42 + 72.7i)17-s + (74 − 128. i)19-s + 31.9·20-s − 80·22-s + (42 − 72.7i)23-s + (30.5 + 52.8i)25-s + (4 − 6.92i)26-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.357 + 0.619i)5-s + 0.353·8-s + (−0.252 − 0.438i)10-s + (0.548 + 0.949i)11-s − 0.0853·13-s + (−0.125 + 0.216i)16-s + (0.599 + 1.03i)17-s + (0.893 − 1.54i)19-s + 0.357·20-s − 0.775·22-s + (0.380 − 0.659i)23-s + (0.244 + 0.422i)25-s + (0.0301 − 0.0522i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.505015403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505015403\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (4 - 6.92i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-20 - 34.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 4T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-42 - 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-74 + 128. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-42 + 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 58T + 2.43e4T^{2} \) |
| 31 | \( 1 + (68 + 117. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-111 + 192. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 420T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + (244 - 422. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-239 - 413. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (274 + 474. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-346 + 599. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-454 - 786. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 524T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-220 - 381. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (608 - 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 684T + 5.71e5T^{2} \) |
| 89 | \( 1 + (302 - 523. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 832T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.714879919780118228268093871157, −9.306199235491868336321694692864, −8.179525701153965263108620433752, −7.32867786215593760669409647592, −6.84522350296149077361290414139, −5.82198346397583932902458569575, −4.75992817625116901059784829602, −3.77550680248969001489138332926, −2.47656822013302420966938165988, −1.00421005397588998533956616212,
0.57707210975312479493743956858, 1.47543416799411403831628700372, 3.06494867622510365688247579511, 3.79315530065622872314601504341, 4.98498115317466576903732732524, 5.85287937243293545413355382586, 7.16423219140379823302192824912, 8.005443474786458595116267818284, 8.712882560227662252369905312791, 9.528140775371864134018027101228
