| L(s) = 1 | + (1.03 − 0.0904i)2-s + (−0.909 + 0.160i)4-s + (−1.76 − 1.36i)5-s + (−2.38 + 1.66i)7-s + (−2.92 + 0.785i)8-s + (−1.95 − 1.25i)10-s + (0.357 + 0.981i)11-s + (−0.102 + 1.17i)13-s + (−2.31 + 1.94i)14-s + (−1.22 + 0.444i)16-s + (−5.46 − 1.46i)17-s + (−3.77 − 2.17i)19-s + (1.82 + 0.962i)20-s + (0.457 + 0.981i)22-s + (−2.49 + 3.55i)23-s + ⋯ |
| L(s) = 1 | + (0.730 − 0.0639i)2-s + (−0.454 + 0.0801i)4-s + (−0.790 − 0.612i)5-s + (−0.900 + 0.630i)7-s + (−1.03 + 0.277i)8-s + (−0.616 − 0.397i)10-s + (0.107 + 0.295i)11-s + (−0.0285 + 0.326i)13-s + (−0.617 + 0.518i)14-s + (−0.305 + 0.111i)16-s + (−1.32 − 0.355i)17-s + (−0.866 − 0.500i)19-s + (0.408 + 0.215i)20-s + (0.0976 + 0.209i)22-s + (−0.519 + 0.742i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0197035 + 0.143727i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0197035 + 0.143727i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.76 + 1.36i)T \) |
| good | 2 | \( 1 + (-1.03 + 0.0904i)T + (1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (2.38 - 1.66i)T + (2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.357 - 0.981i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.102 - 1.17i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (5.46 + 1.46i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (3.77 + 2.17i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.49 - 3.55i)T + (-7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (-5.99 - 5.02i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.88 + 10.6i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.84 + 6.89i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.34 - 2.79i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.986 - 2.11i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-0.209 - 0.298i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (1.16 + 1.16i)T + 53iT^{2} \) |
| 59 | \( 1 + (9.77 + 3.55i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.654 - 3.70i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.570 - 0.0498i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (2.83 - 1.63i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.56 - 13.2i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (2.43 - 2.89i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.416 - 4.75i)T + (-81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (-1.62 + 2.81i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.8 + 5.05i)T + (62.3 + 74.3i)T^{2} \) |
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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
−11.87325484429921549859936138677, −11.11525350141301403175884815517, −9.478700863765400096711660500334, −9.089415640260273559775647730574, −8.126969257512543162540091595697, −6.79829619304813621965715623724, −5.78537880445969994392437029758, −4.61944380965104766401609655309, −3.95618101817467959963281544749, −2.62230441185709511574535922784,
0.07195493021097932933580949234, 2.93861146593157586911579683288, 3.89363519218049766318413133109, 4.64232714706323604948734799920, 6.28325907816315445108471846234, 6.66349495125400573704026382076, 8.099662599197395662573356904940, 8.914593197783857293622473590936, 10.20278123032464287759212125855, 10.72246240394942384831138668243
