| L(s) = 1 | + (−2 − 3.46i)2-s + (−7.99 + 13.8i)4-s + (−9.99 + 17.3i)5-s + (101. + 175. i)7-s + 63.9·8-s + 79.9·10-s + (−10.4 − 18.1i)11-s + (116. − 202. i)13-s + (404. − 700. i)14-s + (−128 − 221. i)16-s − 1.30e3·17-s − 1.42e3·19-s + (−159. − 276. i)20-s + (−41.9 + 72.6i)22-s + (964. − 1.66e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.178 + 0.309i)5-s + (0.780 + 1.35i)7-s + 0.353·8-s + 0.252·10-s + (−0.0261 − 0.0452i)11-s + (0.191 − 0.331i)13-s + (0.551 − 0.955i)14-s + (−0.125 − 0.216i)16-s − 1.09·17-s − 0.908·19-s + (−0.0893 − 0.154i)20-s + (−0.0184 + 0.0320i)22-s + (0.380 − 0.658i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6093259536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6093259536\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 + 3.46i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (9.99 - 17.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-101. - 175. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (10.4 + 18.1i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-116. + 202. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 1.30e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.42e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-964. + 1.66e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (2.18e3 + 3.78e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (4.57e3 - 7.92e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 8.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-7.90e3 + 1.36e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (2.01e3 + 3.49e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.33e4 - 2.31e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 3.78e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (1.48e4 - 2.57e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-8.55e3 - 1.48e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (3.43e4 - 5.95e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 2.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.26e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (3.38e4 + 5.86e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (2.30e4 + 3.99e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + 2.60e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (8.81e3 + 1.52e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−12.22434852139410422410872483077, −11.17363992958646363727854432884, −10.62633048861241117654763685885, −8.989426030256510038006388300023, −8.627087492876320545111044097511, −7.25449739041886831933076644047, −5.78223082267176963693846584019, −4.49785597365734948470280559618, −2.88056401868971062798928724182, −1.77358715059821654217282516560,
0.22073044085337699853641571382, 1.65905289548747462541588511978, 4.01806381857974807478117688819, 4.89916827673981769096616311167, 6.48443882134235017550026311290, 7.43625417667988921440937229007, 8.355614519691513043167325834377, 9.383863448113881228576811112275, 10.69416353525351900464375981660, 11.24215569013173476172919784993
