| L(s) = 1 | + (−2.35 + 1.71i)2-s + (5.78 + 17.8i)3-s + (−7.27 + 22.3i)4-s + (−20.2 − 14.6i)5-s + (−44.0 − 32.0i)6-s + (−64.4 + 198. i)7-s + (−49.9 − 153. i)8-s + (−86.9 + 63.1i)9-s + 72.7·10-s + (350. − 196. i)11-s − 440.·12-s + (76.7 − 55.7i)13-s + (−187. − 577. i)14-s + (144. − 445. i)15-s + (−228. − 165. i)16-s + (−1.31e3 − 955. i)17-s + ⋯ |
| L(s) = 1 | + (−0.416 + 0.302i)2-s + (0.371 + 1.14i)3-s + (−0.227 + 0.699i)4-s + (−0.361 − 0.262i)5-s + (−0.499 − 0.363i)6-s + (−0.496 + 1.52i)7-s + (−0.275 − 0.849i)8-s + (−0.357 + 0.260i)9-s + 0.230·10-s + (0.872 − 0.489i)11-s − 0.883·12-s + (0.125 − 0.0915i)13-s + (−0.255 − 0.787i)14-s + (0.165 − 0.510i)15-s + (−0.223 − 0.162i)16-s + (−1.10 − 0.802i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.205862 - 0.828134i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.205862 - 0.828134i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (20.2 + 14.6i)T \) |
| 11 | \( 1 + (-350. + 196. i)T \) |
| good | 2 | \( 1 + (2.35 - 1.71i)T + (9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (-5.78 - 17.8i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + (64.4 - 198. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-76.7 + 55.7i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.31e3 + 955. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-523. - 1.61e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + 3.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-1.25e3 + 3.84e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (496. - 360. i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (4.79e3 - 1.47e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.91e3 - 5.89e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 8.44e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-5.53e3 - 1.70e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (2.36e4 - 1.71e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.75e3 - 5.41e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.85e4 + 1.34e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 - 2.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (2.39e3 + 1.74e3i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.34e4 + 4.14e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (6.41e4 - 4.65e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-9.39e3 - 6.82e3i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 4.49e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (7.56e4 - 5.49e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−15.58177854220774196793732103707, −14.03085373033068190606522691108, −12.47740644382925629466352429245, −11.65446479736503915977098098983, −9.714321084660238561745306295518, −9.063225708525474152715136102486, −8.190023445923227162563972360362, −6.24087900252698105562142705225, −4.35925782980938585607574594246, −3.11718432945659083982043984733,
0.46459168852465630421790184393, 1.85486532754607616109109546331, 4.12925241793688315246998446754, 6.52407224270137730833059138423, 7.34305248293186067971551202366, 8.824606405750929628490154448744, 10.17879543464504680152635792753, 11.16158311303621546890561205469, 12.64390420833605397360457177367, 13.75234314751796664248266483572
