| L(s) = 1 | + (1.67 − 0.969i)3-s + (−72.6 − 125. i)5-s + 621.·7-s + (−362. + 628. i)9-s + 554.·11-s + (−3.12e3 − 1.80e3i)13-s + (−244. − 140. i)15-s + (−2.80e3 − 4.86e3i)17-s + (4.47e3 − 5.20e3i)19-s + (1.04e3 − 602. i)21-s + (2.88e3 − 4.99e3i)23-s + (−2.75e3 + 4.76e3i)25-s + 2.81e3i·27-s + (−1.34e4 − 7.77e3i)29-s − 4.33e4i·31-s + ⋯ |
| L(s) = 1 | + (0.0621 − 0.0359i)3-s + (−0.581 − 1.00i)5-s + 1.81·7-s + (−0.497 + 0.861i)9-s + 0.416·11-s + (−1.42 − 0.821i)13-s + (−0.0723 − 0.0417i)15-s + (−0.571 − 0.990i)17-s + (0.651 − 0.758i)19-s + (0.112 − 0.0651i)21-s + (0.236 − 0.410i)23-s + (−0.176 + 0.304i)25-s + 0.143i·27-s + (−0.551 − 0.318i)29-s − 1.45i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.905927 - 1.20839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.905927 - 1.20839i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-4.47e3 + 5.20e3i)T \) |
| good | 3 | \( 1 + (-1.67 + 0.969i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (72.6 + 125. i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 - 621.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 554.T + 1.77e6T^{2} \) |
| 13 | \( 1 + (3.12e3 + 1.80e3i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (2.80e3 + 4.86e3i)T + (-1.20e7 + 2.09e7i)T^{2} \) |
| 23 | \( 1 + (-2.88e3 + 4.99e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.34e4 + 7.77e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + 4.33e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 5.29e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (5.07e4 - 2.92e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-1.99e4 - 3.46e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (8.27e4 - 1.43e5i)T + (-5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (2.29e5 + 1.32e5i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.05e5 + 6.09e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-4.71e4 + 8.16e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.96e5 - 1.13e5i)T + (4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (1.48e5 - 8.59e4i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-1.93e5 - 3.35e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-7.11e4 + 4.10e4i)T + (1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 1.10e6T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.03e6 - 6.00e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-6.88e5 + 3.97e5i)T + (4.16e11 - 7.21e11i)T^{2} \) |
| 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
−12.92702903300102666660695789439, −11.68227489131164745786777281582, −11.13707289029593084978455853174, −9.374995073608110859551587546422, −8.146847803498225685819052264418, −7.57521498711404991711856521428, −5.11410150207703446039734972470, −4.69245500223054640508818228917, −2.28940875075613432738622615698, −0.57185899386372993101540557460,
1.75790941902490299817907776420, 3.54729631956046227693430129224, 4.97363627026551078367800466009, 6.73200902051292362156503897588, 7.77429116888438630418987779410, 8.961324233266507752600616292082, 10.49844196409751795369432260662, 11.57079107387121791253699929851, 12.04654337974620207568981765066, 14.11625314937497487099072344331
