L(s) = 1 | + (0.642 + 0.766i)2-s + (−0.811 + 2.22i)3-s + (−0.173 + 0.984i)4-s + (−2.22 + 0.811i)6-s + (2.73 + 1.57i)7-s + (−0.866 + 0.500i)8-s + (−2.00 − 1.68i)9-s + (−0.688 − 1.19i)11-s + (−2.05 − 1.18i)12-s + (1.47 + 4.06i)13-s + (0.547 + 3.10i)14-s + (−0.939 − 0.342i)16-s + (0.833 + 0.993i)17-s − 2.62i·18-s + (−1.08 + 4.22i)19-s + ⋯ |
L(s) = 1 | + (0.454 + 0.541i)2-s + (−0.468 + 1.28i)3-s + (−0.0868 + 0.492i)4-s + (−0.909 + 0.331i)6-s + (1.03 + 0.596i)7-s + (−0.306 + 0.176i)8-s + (−0.669 − 0.562i)9-s + (−0.207 − 0.359i)11-s + (−0.592 − 0.342i)12-s + (0.410 + 1.12i)13-s + (0.146 + 0.830i)14-s + (−0.234 − 0.0855i)16-s + (0.202 + 0.240i)17-s − 0.618i·18-s + (−0.249 + 0.968i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0183021 - 1.70739i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0183021 - 1.70739i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.642 - 0.766i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (1.08 - 4.22i)T \) |
good | 3 | \( 1 + (0.811 - 2.22i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-2.73 - 1.57i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.688 + 1.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.47 - 4.06i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.833 - 0.993i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (2.09 + 0.369i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.0998 - 0.0837i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.173 - 0.300i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.3iT - 37T^{2} \) |
| 41 | \( 1 + (10.3 + 3.76i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-11.3 + 2.00i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.0491 + 0.0585i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-6.12 - 1.08i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.27 + 5.26i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.36 + 7.72i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.662 - 0.789i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.02 - 11.4i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (4.18 - 11.5i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-13.1 - 4.79i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (9.85 + 5.68i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (17.1 - 6.23i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-10.4 - 12.4i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−10.57151031273383213040306239934, −9.592687169368395106227666245167, −8.735275980655945073340967049703, −8.109971800749068333025669144101, −6.92513098045934378156665946264, −5.70548649714747563744377008191, −5.38641098007420116916073852631, −4.26371560223382671051345826319, −3.77692896333477306376529122431, −2.06016639593314090983344110352,
0.75834218491791096307268739757, 1.72323931758465883734404084624, 2.93539272275797079523742478934, 4.36092889854761056525889646921, 5.23074643669932872851084253074, 6.13568768624196230510195987959, 7.09049141749412860464115814144, 7.78120863038892750049752202106, 8.559643118606487505052651276810, 9.938656659681365012459395578775