L(s) = 1 | + (−1.41 − 0.0402i)2-s + (1.99 + 0.113i)4-s + (−1.51 − 0.626i)5-s + (−1.32 − 1.32i)7-s + (−2.81 − 0.241i)8-s + (2.11 + 0.946i)10-s + (−0.938 + 2.26i)11-s + (−4.73 + 1.96i)13-s + (1.82 + 1.92i)14-s + (3.97 + 0.454i)16-s − 5.78i·17-s + (−1.04 + 0.431i)19-s + (−2.94 − 1.42i)20-s + (1.41 − 3.16i)22-s + (−4.29 + 4.29i)23-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.0284i)2-s + (0.998 + 0.0568i)4-s + (−0.676 − 0.280i)5-s + (−0.500 − 0.500i)7-s + (−0.996 − 0.0852i)8-s + (0.668 + 0.299i)10-s + (−0.283 + 0.683i)11-s + (−1.31 + 0.543i)13-s + (0.486 + 0.514i)14-s + (0.993 + 0.113i)16-s − 1.40i·17-s + (−0.239 + 0.0990i)19-s + (−0.659 − 0.318i)20-s + (0.302 − 0.675i)22-s + (−0.895 + 0.895i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00226447 - 0.0836110i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00226447 - 0.0836110i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 + 0.0402i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.51 + 0.626i)T + (3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (1.32 + 1.32i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.938 - 2.26i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (4.73 - 1.96i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 5.78iT - 17T^{2} \) |
| 19 | \( 1 + (1.04 - 0.431i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (4.29 - 4.29i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.389 - 0.940i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + (3.67 + 1.52i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (0.474 - 0.474i)T - 41iT^{2} \) |
| 43 | \( 1 + (0.409 - 0.987i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 2.73iT - 47T^{2} \) |
| 53 | \( 1 + (-4.55 + 10.9i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-8.68 - 3.59i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.48 - 3.58i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-6.11 - 14.7i)T + (-47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-10.5 - 10.5i)T + 71iT^{2} \) |
| 73 | \( 1 + (-6.86 + 6.86i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.4iT - 79T^{2} \) |
| 83 | \( 1 + (15.2 - 6.30i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (6.07 + 6.07i)T + 89iT^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−11.42455522193260122777346917177, −10.01794290926889482392443604411, −9.693100047347618951175532115847, −8.489398286984066883700278897115, −7.31392750532836793983200738938, −7.04822930548404483134164562082, −5.30142375754404367276997564106, −3.84209510818853128991652009565, −2.25570209569323028055649831549, −0.07831817114782933634809844981,
2.36048144184188977126281203952, 3.62951942989377708814909721047, 5.56108375777577510891419666382, 6.57071109484550210854982741260, 7.70589116642229818643575236241, 8.360146069546140324533040451808, 9.437318892751923508162805209901, 10.39391147011393725031682709187, 11.08464286069668139570275316210, 12.23255719005124361350238284625