| L(s) = 1 | + (0.964 + 1.03i)2-s + (0.554 + 0.722i)3-s + (−0.139 + 1.99i)4-s + (0.772 + 0.592i)5-s + (−0.212 + 1.26i)6-s + (2.49 − 0.891i)7-s + (−2.19 + 1.78i)8-s + (0.561 − 2.09i)9-s + (0.132 + 1.37i)10-s + (−3.25 − 0.428i)11-s + (−1.51 + 1.00i)12-s + (−4.64 − 1.92i)13-s + (3.32 + 1.71i)14-s + 0.886i·15-s + (−3.96 − 0.554i)16-s + (3.80 + 2.19i)17-s + ⋯ |
| L(s) = 1 | + (0.682 + 0.731i)2-s + (0.320 + 0.417i)3-s + (−0.0695 + 0.997i)4-s + (0.345 + 0.264i)5-s + (−0.0867 + 0.518i)6-s + (0.941 − 0.337i)7-s + (−0.776 + 0.629i)8-s + (0.187 − 0.699i)9-s + (0.0417 + 0.433i)10-s + (−0.982 − 0.129i)11-s + (−0.438 + 0.290i)12-s + (−1.28 − 0.533i)13-s + (0.888 + 0.458i)14-s + 0.228i·15-s + (−0.990 − 0.138i)16-s + (0.921 + 0.532i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0897 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0897 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42990 + 1.30684i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42990 + 1.30684i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.964 - 1.03i)T \) |
| 7 | \( 1 + (-2.49 + 0.891i)T \) |
| good | 3 | \( 1 + (-0.554 - 0.722i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (-0.772 - 0.592i)T + (1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (3.25 + 0.428i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (4.64 + 1.92i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (-3.80 - 2.19i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.313 - 2.38i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (1.12 - 4.21i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.82 + 9.23i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-0.642 + 1.11i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.26 + 2.50i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (-4.76 - 4.76i)T + 41iT^{2} \) |
| 43 | \( 1 + (0.0816 + 0.197i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (5.27 - 3.04i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.41 + 0.844i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (1.84 - 14.0i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (7.84 - 1.03i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-4.59 - 5.98i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (-6.86 + 6.86i)T - 71iT^{2} \) |
| 73 | \( 1 + (-9.46 + 2.53i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (6.16 - 3.56i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.32 - 1.79i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-7.96 - 2.13i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−12.54643043269646391687899717513, −11.81651557861617110453756797960, −10.40524994316988896963859189911, −9.619590857924990482167219780943, −8.051633665787009072435760190793, −7.68861336112144036318647279642, −6.18338242885071771634721666875, −5.15976750495678281683320915056, −4.06203341857613610593257723879, −2.70078029338222087305293342878,
1.78173591637930089624515762803, 2.79771041684807355233950965677, 4.97611392191706454717491509097, 5.09110468456966997909273614705, 6.96943215059874585649579564281, 8.038128273125078206738921070979, 9.266541954939489839265925162475, 10.29726462020507996147510454083, 11.14939794454463873838733542944, 12.30689345380477136546485324536
