| 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 | \( 1 + (10.8 - 5.05i)T + (62.3 - 74.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.72246240394942384831138668243, −10.20278123032464287759212125855, −8.914593197783857293622473590936, −8.099662599197395662573356904940, −6.66349495125400573704026382076, −6.28325907816315445108471846234, −4.64232714706323604948734799920, −3.89363519218049766318413133109, −2.93861146593157586911579683288, −0.07195493021097932933580949234,
2.62230441185709511574535922784, 3.95618101817467959963281544749, 4.61944380965104766401609655309, 5.78537880445969994392437029758, 6.79829619304813621965715623724, 8.126969257512543162540091595697, 9.089415640260273559775647730574, 9.478700863765400096711660500334, 11.11525350141301403175884815517, 11.87325484429921549859936138677
