| L(s) = 1 | + (−32.0 + 18.5i)3-s + (−143. − 82.8i)5-s + (197. − 280. i)7-s + (321. − 556. i)9-s + (−86.5 − 149. i)11-s − 1.96e3i·13-s + 6.14e3·15-s + (3.19e3 − 1.84e3i)17-s + (−3.95e3 − 2.28e3i)19-s + (−1.13e3 + 1.26e4i)21-s + (−7.89e3 + 1.36e4i)23-s + (5.92e3 + 1.02e4i)25-s − 3.18e3i·27-s − 2.37e4·29-s + (−1.78e3 + 1.03e3i)31-s + ⋯ |
| L(s) = 1 | + (−1.18 + 0.685i)3-s + (−1.14 − 0.663i)5-s + (0.575 − 0.818i)7-s + (0.440 − 0.763i)9-s + (−0.0650 − 0.112i)11-s − 0.893i·13-s + 1.81·15-s + (0.651 − 0.375i)17-s + (−0.576 − 0.332i)19-s + (−0.122 + 1.36i)21-s + (−0.648 + 1.12i)23-s + (0.379 + 0.657i)25-s − 0.161i·27-s − 0.975·29-s + (−0.0599 + 0.0346i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3142239246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3142239246\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-197. + 280. i)T \) |
| good | 3 | \( 1 + (32.0 - 18.5i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (143. + 82.8i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (86.5 + 149. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + 1.96e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-3.19e3 + 1.84e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (3.95e3 + 2.28e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (7.89e3 - 1.36e4i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.37e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (1.78e3 - 1.03e3i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (2.42e4 - 4.20e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 2.64e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.84e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.21e5 - 7.01e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.27e5 - 2.20e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-8.41e4 + 4.86e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.50e4 - 8.69e3i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (6.03e4 + 1.04e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.39e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-9.64e4 + 5.56e4i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.07e5 - 5.33e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 3.83e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (6.68e5 + 3.85e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 2.92e5iT - 8.32e11T^{2} \) |
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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
−12.42900360763405483859115015001, −11.57263288474549297134616430481, −10.88717467888285949786733571725, −9.893862815480903657763299383985, −8.285006727319587064731050552086, −7.38574589194846902273853767936, −5.64632354973191442383530992300, −4.70485298776498473567930159414, −3.74207243203723888169485832991, −0.898939335997287904312654808644,
0.16584968208096145902304097338, 1.97101918105287149752957402195, 3.94479473071691667050746170054, 5.43860354190689549242123849936, 6.55403971104233678823628696599, 7.52514705588714917937222307659, 8.653299481494355503398906871520, 10.46993714171426462362528892390, 11.44131208632531223241362773890, 11.93952207752930455907050375786
