| L(s) = 1 | + 5.65·2-s + 32.0·4-s + 484i·7-s + 181.·8-s + 1.34e3i·11-s + 3.36e3i·13-s + 2.73e3i·14-s + 1.02e3·16-s + 12.7·17-s − 5.74e3·19-s + 7.58e3i·22-s + 3.37e3·23-s + 1.90e4i·26-s + 1.54e4i·28-s − 2.93e4i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 1.41i·7-s + 0.353·8-s + 1.00i·11-s + 1.53i·13-s + 0.997i·14-s + 0.250·16-s + 0.00259·17-s − 0.837·19-s + 0.712i·22-s + 0.277·23-s + 1.08i·26-s + 0.705i·28-s − 1.20i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.876867661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876867661\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 484iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.34e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 3.36e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 12.7T + 2.41e7T^{2} \) |
| 19 | \( 1 + 5.74e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 3.37e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 2.93e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.97e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 5.25e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 3.70e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 3.80e3iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 7.67e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.38e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.49e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.32e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.68e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 5.31e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.36e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 3.51e4T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.09e4T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.21e5iT - 8.32e11T^{2} \) |
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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
−10.73735332202647377351964786089, −9.436121976761383214248962553173, −8.936170415899681992909591643885, −7.63194829668230816022765872305, −6.64080699897648056910643126281, −5.81521161170613892319151878775, −4.78776845106633537829138911965, −3.91474859212138400031922609444, −2.38796834328966200326328669882, −1.86688445988452093051996070735,
0.28804060155400237694847609151, 1.33128022184041097542168907228, 3.01579393317100391610220840970, 3.70836332442545964810838426388, 4.84508188244835724964743012338, 5.81795119501360122838768672150, 6.82987913097687210742662742186, 7.71075567814694647744081727969, 8.571759245011409865072805521434, 9.997350352819748807530890659927
