| L(s) = 1 | + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (190. − 109. i)5-s + (−180. + 313. i)7-s − 181. i·8-s − 1.24e3·10-s + (56.7 + 32.7i)11-s + (1.62e3 + 2.81e3i)13-s + (1.77e3 − 1.02e3i)14-s + (−512. + 886. i)16-s + 6.32e3i·17-s − 8.69e3·19-s + (6.09e3 + 3.51e3i)20-s + (−185. − 321. i)22-s + (−1.13e4 + 6.54e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (1.52 − 0.879i)5-s + (−0.527 + 0.913i)7-s − 0.353i·8-s − 1.24·10-s + (0.0426 + 0.0246i)11-s + (0.739 + 1.28i)13-s + (0.645 − 0.372i)14-s + (−0.125 + 0.216i)16-s + 1.28i·17-s − 1.26·19-s + (0.761 + 0.439i)20-s + (−0.0174 − 0.0301i)22-s + (−0.932 + 0.538i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.270223508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.270223508\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 + 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-190. + 109. i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (180. - 313. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-56.7 - 32.7i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-1.62e3 - 2.81e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 6.32e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 8.69e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.13e4 - 6.54e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (6.57e3 + 3.79e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.88e4 + 3.26e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 2.14e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-1.78e4 + 1.03e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (6.12e4 - 1.06e5i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.36e4 + 7.88e3i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 2.41e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-6.44e4 + 3.71e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (9.68e4 - 1.67e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-6.78e4 - 1.17e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 2.62e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.18e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-3.41e5 + 5.90e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (5.14e5 + 2.97e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 6.52e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (8.77e5 - 1.51e6i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−12.06386691329413199692037521089, −10.78723782202207388168637986757, −9.674769600412065045451362842962, −9.104747174127751688449538517683, −8.333414265477898897854918530532, −6.36048998685115051398935532232, −5.84322858552710448479319713119, −4.13262461719487822607315591648, −2.21852094005953603881247864648, −1.54046493613576078617673146730,
0.42357019529693825410030169697, 1.99000285295231285330517753250, 3.32493854493522359238758080689, 5.37178400815123863379768560159, 6.41564945995339540782206038824, 7.08090753404833566025017227348, 8.500610838643259687950460559464, 9.726124678000054544102752144617, 10.35801073713278638301568437729, 10.95370206143282055377005461191
