L(s) = 1 | + (−0.965 + 0.258i)3-s + (−0.5 − 0.866i)4-s + (−0.448 + 0.258i)7-s + (0.866 − 0.499i)9-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.499 + 0.866i)16-s + (−1.86 − 0.5i)19-s + (0.366 − 0.366i)21-s + (−0.707 + 0.707i)27-s + (0.448 + 0.258i)28-s + (−1 − i)31-s + (−0.866 − 0.5i)36-s + (−1.22 − 0.707i)37-s + (0.866 + 0.500i)39-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)3-s + (−0.5 − 0.866i)4-s + (−0.448 + 0.258i)7-s + (0.866 − 0.499i)9-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.499 + 0.866i)16-s + (−1.86 − 0.5i)19-s + (0.366 − 0.366i)21-s + (−0.707 + 0.707i)27-s + (0.448 + 0.258i)28-s + (−1 − i)31-s + (−0.866 − 0.5i)36-s + (−1.22 − 0.707i)37-s + (0.866 + 0.500i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2224220517\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2224220517\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (1 + i)T + iT^{2} \) |
| 37 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 - 1.93T + T^{2} \) |
| 79 | \( 1 - 1.73iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.869925839390100053556356908485, −9.330881987264714266549613145288, −8.331076592125628208659127566127, −7.00655344955167514311810134821, −6.26967277577634171569013439037, −5.48172168487979534572861317056, −4.77630670119967318378677245032, −3.78971912271402120139586310613, −2.06310941971067201955068874869, −0.22888063412094698291290875462,
2.01066125703874419260014822840, 3.56688904395784709297476869362, 4.45438172651870243624124860761, 5.26476153063917705151307982981, 6.58960122946672671106058247119, 6.97387185160634084362339120813, 8.039999090317525187930308162107, 8.848890163716550086136392129615, 9.853919137239872863053128596207, 10.55376403790792924731089337669