L(s) = 1 | + (1 − i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + (−4.90 + 0.988i)5-s − 2.44·6-s + (2.14 − 2.14i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−3.91 + 5.89i)10-s − 12.9·11-s + (−2.44 + 2.44i)12-s + (10.2 + 10.2i)13-s − 4.29i·14-s + (7.21 + 4.79i)15-s − 4·16-s + (−2.98 + 2.98i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.980 + 0.197i)5-s − 0.408·6-s + (0.307 − 0.307i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.391 + 0.589i)10-s − 1.18·11-s + (−0.204 + 0.204i)12-s + (0.788 + 0.788i)13-s − 0.307i·14-s + (0.480 + 0.319i)15-s − 0.250·16-s + (−0.175 + 0.175i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.181742001\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.181742001\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 + (4.90 - 0.988i)T \) |
| 23 | \( 1 + (-3.39 - 3.39i)T \) |
good | 7 | \( 1 + (-2.14 + 2.14i)T - 49iT^{2} \) |
| 11 | \( 1 + 12.9T + 121T^{2} \) |
| 13 | \( 1 + (-10.2 - 10.2i)T + 169iT^{2} \) |
| 17 | \( 1 + (2.98 - 2.98i)T - 289iT^{2} \) |
| 19 | \( 1 - 26.4iT - 361T^{2} \) |
| 29 | \( 1 + 45.6iT - 841T^{2} \) |
| 31 | \( 1 - 37.4T + 961T^{2} \) |
| 37 | \( 1 + (44.9 - 44.9i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 39.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-29.7 - 29.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (10.2 - 10.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-34.6 - 34.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 22.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-19.6 + 19.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 131.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-49.0 - 49.0i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 59.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (31.0 + 31.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 169. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-8.17 + 8.17i)T - 9.40e3iT^{2} \) |
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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
−10.72139686836353864621455633694, −9.751771808648753134758281058379, −8.274427657654910192474273181825, −7.82624311415203701430602044901, −6.67803540909435447086084707302, −5.81131891706583080783204511108, −4.62734551214964500488800647413, −3.88225654624906871989308355013, −2.60554862120441367480029342155, −1.15707646936634119575460002253,
0.43392932404901623490299174244, 2.78153096761927388357036821660, 3.80357819651536314130525685039, 4.97524307264127032425251980930, 5.33914522339570580116389434783, 6.66719833632297643338239295752, 7.53973989256002761419325140128, 8.410779457350918817629942001148, 9.048081628763746263368557476088, 10.60469520383933469517849673670