| L(s) = 1 | + (−0.813 + 1.15i)2-s + (2.77 − 1.06i)3-s + (−0.677 − 1.88i)4-s + (2.43 − 3.12i)5-s + (−1.01 + 4.07i)6-s + (−0.0546 + 0.152i)7-s + (2.72 + 0.747i)8-s + (4.31 − 3.90i)9-s + (1.62 + 5.35i)10-s + (0.865 + 3.85i)11-s + (−3.88 − 4.49i)12-s + (−3.86 + 1.06i)13-s + (−0.132 − 0.187i)14-s + (3.41 − 11.2i)15-s + (−3.08 + 2.54i)16-s + (−4.28 + 1.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.575 + 0.818i)2-s + (1.59 − 0.617i)3-s + (−0.338 − 0.940i)4-s + (1.08 − 1.39i)5-s + (−0.415 + 1.66i)6-s + (−0.0206 + 0.0577i)7-s + (0.964 + 0.264i)8-s + (1.43 − 1.30i)9-s + (0.515 + 1.69i)10-s + (0.260 + 1.16i)11-s + (−1.12 − 1.29i)12-s + (−1.07 + 0.296i)13-s + (−0.0353 − 0.0501i)14-s + (0.881 − 2.90i)15-s + (−0.770 + 0.637i)16-s + (−1.03 + 0.315i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.00037 - 0.326321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00037 - 0.326321i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.813 - 1.15i)T \) |
| good | 3 | \( 1 + (-2.77 + 1.06i)T + (2.22 - 2.01i)T^{2} \) |
| 5 | \( 1 + (-2.43 + 3.12i)T + (-1.21 - 4.85i)T^{2} \) |
| 7 | \( 1 + (0.0546 - 0.152i)T + (-5.41 - 4.44i)T^{2} \) |
| 11 | \( 1 + (-0.865 - 3.85i)T + (-9.94 + 4.70i)T^{2} \) |
| 13 | \( 1 + (3.86 - 1.06i)T + (11.1 - 6.68i)T^{2} \) |
| 17 | \( 1 + (4.28 - 1.29i)T + (14.1 - 9.44i)T^{2} \) |
| 19 | \( 1 + (-1.72 - 3.42i)T + (-11.3 + 15.2i)T^{2} \) |
| 23 | \( 1 + (-4.90 + 0.727i)T + (22.0 - 6.67i)T^{2} \) |
| 29 | \( 1 + (0.640 + 3.69i)T + (-27.3 + 9.76i)T^{2} \) |
| 31 | \( 1 + (0.986 - 1.47i)T + (-11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (3.25 - 2.80i)T + (5.42 - 36.5i)T^{2} \) |
| 41 | \( 1 + (2.26 - 9.04i)T + (-36.1 - 19.3i)T^{2} \) |
| 43 | \( 1 + (4.89 + 11.0i)T + (-28.8 + 31.8i)T^{2} \) |
| 47 | \( 1 + (6.16 + 7.50i)T + (-9.16 + 46.0i)T^{2} \) |
| 53 | \( 1 + (0.0866 - 0.499i)T + (-49.9 - 17.8i)T^{2} \) |
| 59 | \( 1 + (2.03 - 7.36i)T + (-50.6 - 30.3i)T^{2} \) |
| 61 | \( 1 + (-4.10 - 0.100i)T + (60.9 + 2.99i)T^{2} \) |
| 67 | \( 1 + (2.19 - 2.08i)T + (3.28 - 66.9i)T^{2} \) |
| 71 | \( 1 + (-3.42 + 0.168i)T + (70.6 - 6.95i)T^{2} \) |
| 73 | \( 1 + (-3.75 - 10.4i)T + (-56.4 + 46.3i)T^{2} \) |
| 79 | \( 1 + (-6.39 + 0.629i)T + (77.4 - 15.4i)T^{2} \) |
| 83 | \( 1 + (3.20 + 2.76i)T + (12.1 + 82.1i)T^{2} \) |
| 89 | \( 1 + (17.2 + 2.56i)T + (85.1 + 25.8i)T^{2} \) |
| 97 | \( 1 + (-13.6 + 2.71i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.02820278348629647562694838306, −9.660138446092209861246529554177, −8.870969679055453255099211853056, −8.416073208789051423670044543555, −7.34204418857873611542384849771, −6.63162160044172537979317815268, −5.23869850673413192688513315735, −4.35436438693042471369061313228, −2.21230023772651139528139830794, −1.51167793397084682542185230619,
2.09973298339907529213977560361, 2.90494209172822115669805181658, 3.43722715793129506080734346259, 4.96552807067191938760315360070, 6.76600719504453004548584281017, 7.52181575004639907416755847561, 8.691735576300366878035504820453, 9.331166477183620681813805020100, 9.874663801182020085538628120238, 10.78021074332700115191451252300
