| L(s) = 1 | + (−1.98 − 1.98i)2-s + (2.72 + 4.72i)3-s + 3.88i·4-s + (0.225 + 0.390i)5-s + (3.96 − 14.8i)6-s + (−0.538 − 0.932i)7-s + (−0.218 + 0.218i)8-s + (−10.3 + 17.9i)9-s + (0.327 − 1.22i)10-s + (4.03 + 15.0i)11-s + (−18.3 + 10.6i)12-s + (−1.21 − 0.325i)13-s + (−0.782 + 2.92i)14-s + (−1.22 + 2.12i)15-s + 16.4·16-s + (−5.88 − 5.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.993 − 0.993i)2-s + (0.909 + 1.57i)3-s + 0.972i·4-s + (0.0450 + 0.0780i)5-s + (0.661 − 2.46i)6-s + (−0.0769 − 0.133i)7-s + (−0.0273 + 0.0273i)8-s + (−1.15 + 1.99i)9-s + (0.0327 − 0.122i)10-s + (0.366 + 1.36i)11-s + (−1.53 + 0.884i)12-s + (−0.0934 − 0.0250i)13-s + (−0.0559 + 0.208i)14-s + (−0.0819 + 0.141i)15-s + 1.02·16-s + (−0.345 − 0.345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.880381 + 0.468635i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.880381 + 0.468635i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (-39.4 - 101. i)T \) |
| good | 2 | \( 1 + (1.98 + 1.98i)T + 4iT^{2} \) |
| 3 | \( 1 + (-2.72 - 4.72i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-0.225 - 0.390i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (0.538 + 0.932i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.03 - 15.0i)T + (-104. + 60.5i)T^{2} \) |
| 13 | \( 1 + (1.21 + 0.325i)T + (146. + 84.5i)T^{2} \) |
| 17 | \( 1 + (5.88 + 5.88i)T + 289iT^{2} \) |
| 19 | \( 1 + (-0.820 + 0.820i)T - 361iT^{2} \) |
| 23 | \( 1 + (-28.9 - 28.9i)T + 529iT^{2} \) |
| 29 | \( 1 + (17.7 - 10.2i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (0.274 + 0.158i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-39.7 + 10.6i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (36.0 + 36.0i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + 55.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-54.0 + 14.4i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (20.1 - 75.2i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (18.5 + 69.1i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (49.7 - 28.7i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-19.7 - 73.8i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 97.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-61.4 + 106. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-18.5 - 69.1i)T + (-5.40e3 + 3.12e3i)T^{2} \) |
| 83 | \( 1 + (-78.6 + 45.4i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-56.4 - 97.7i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (2.11 + 3.66i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
| 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
−13.72294655559092903712327962126, −12.18906116331864261018647079918, −10.96125743807153705101857442141, −10.26796120573368141288948377243, −9.361814795945082749763774337929, −8.946699567381281491548908012418, −7.49777071862562244583593385297, −5.01656272362271592435665208026, −3.65008556907655884158621179429, −2.31209685999721147606015283424,
0.970152221811805787324257649195, 3.04581733780019891121840274358, 6.06914805676440472072423975483, 6.80853602924014621953958599491, 7.88316664336992092961485727127, 8.628358226295288406152123227605, 9.302592026063494760875780866614, 11.20124769275060783466529071067, 12.59058465622730745520213064013, 13.32945023449405223523244138554
