L(s) = 1 | + (−0.163 + 1.40i)2-s + (−1.62 + 0.940i)3-s + (−1.94 − 0.458i)4-s + (−1.68 + 1.47i)5-s + (−1.05 − 2.44i)6-s + (0.603 − 2.57i)7-s + (0.960 − 2.66i)8-s + (0.267 − 0.463i)9-s + (−1.79 − 2.60i)10-s + (−1.20 + 0.692i)11-s + (3.60 − 1.08i)12-s − 5.79·13-s + (3.52 + 1.26i)14-s + (1.36 − 3.97i)15-s + (3.58 + 1.78i)16-s + (2.33 + 4.04i)17-s + ⋯ |
L(s) = 1 | + (−0.115 + 0.993i)2-s + (−0.940 + 0.542i)3-s + (−0.973 − 0.229i)4-s + (−0.753 + 0.657i)5-s + (−0.430 − 0.996i)6-s + (0.228 − 0.973i)7-s + (0.339 − 0.940i)8-s + (0.0891 − 0.154i)9-s + (−0.566 − 0.824i)10-s + (−0.361 + 0.208i)11-s + (1.03 − 0.313i)12-s − 1.60·13-s + (0.940 + 0.338i)14-s + (0.351 − 1.02i)15-s + (0.895 + 0.445i)16-s + (0.565 + 0.980i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0961570 - 0.284031i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0961570 - 0.284031i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.163 - 1.40i)T \) |
| 5 | \( 1 + (1.68 - 1.47i)T \) |
| 7 | \( 1 + (-0.603 + 2.57i)T \) |
good | 3 | \( 1 + (1.62 - 0.940i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.20 - 0.692i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.79T + 13T^{2} \) |
| 17 | \( 1 + (-2.33 - 4.04i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.74 - 4.75i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.32 - 4.02i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.04T + 29T^{2} \) |
| 31 | \( 1 + (-0.832 - 1.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.247 + 0.142i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.21iT - 41T^{2} \) |
| 43 | \( 1 - 3.30T + 43T^{2} \) |
| 47 | \( 1 + (-0.119 - 0.0690i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.82 + 3.36i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.832 + 1.44i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.49 + 1.44i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.82 - 10.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.29iT - 71T^{2} \) |
| 73 | \( 1 + (2.33 + 4.04i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.88 + 5.12i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.0814iT - 83T^{2} \) |
| 89 | \( 1 + (3.22 + 1.85i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.88T + 97T^{2} \) |
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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
−14.21802440918870075896184970491, −12.74298367178618259419816217533, −11.66436506624502845738388595232, −10.28268665595349689290749637240, −10.15063180184454835613457387543, −8.061226527930424063280730753240, −7.43266633089005818259954943664, −6.19978364568557463645230798535, −4.92926293196669225164354649440, −3.95312121420654334453369607135,
0.34611432059018200139288731262, 2.59749333982331659422497430426, 4.66556537509150262193044323046, 5.47594071716603713994669607598, 7.29847986605777983098649073438, 8.517370156257241018997905248101, 9.476795974288872478339565149266, 10.87589136140804995048744382191, 11.90271562306414471798328984945, 12.15660441897745849636036748986