L(s) = 1 | + (−2 + 2i)2-s + (−5.15 − 5.15i)3-s − 8i·4-s + (−17.4 − 17.9i)5-s + 20.6·6-s + (−7.30 + 7.30i)7-s + (16 + 16i)8-s − 27.9i·9-s + (70.7 + 1.02i)10-s − 194.·11-s + (−41.2 + 41.2i)12-s + (−142. − 142. i)13-s − 29.2i·14-s + (−2.63 + 182. i)15-s − 64·16-s + (281. − 281. i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (−0.572 − 0.572i)3-s − 0.5i·4-s + (−0.696 − 0.717i)5-s + 0.572·6-s + (−0.149 + 0.149i)7-s + (0.250 + 0.250i)8-s − 0.344i·9-s + (0.707 + 0.0102i)10-s − 1.60·11-s + (−0.286 + 0.286i)12-s + (−0.841 − 0.841i)13-s − 0.149i·14-s + (−0.0117 + 0.809i)15-s − 0.250·16-s + (0.974 − 0.974i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.02962500124\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02962500124\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 - 2i)T \) |
| 5 | \( 1 + (17.4 + 17.9i)T \) |
| 23 | \( 1 + (77.9 + 77.9i)T \) |
good | 3 | \( 1 + (5.15 + 5.15i)T + 81iT^{2} \) |
| 7 | \( 1 + (7.30 - 7.30i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 + 194.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (142. + 142. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-281. + 281. i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + 479. iT - 1.30e5T^{2} \) |
| 29 | \( 1 - 968. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 293.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (256. - 256. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 2.02e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (2.24e3 + 2.24e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (-478. + 478. i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (1.37e3 + 1.37e3i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 - 5.40e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.14e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (3.54e3 - 3.54e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 1.70e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (76.1 + 76.1i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 89.8iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-4.48e3 - 4.48e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.45e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (1.54e3 - 1.54e3i)T - 8.85e7iT^{2} \) |
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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
−10.85599813310823085946294320554, −9.784663325084023044593615289881, −8.719940934962930341355807609815, −7.63816385545832246208243534224, −7.11429200289806214183414399331, −5.54607339634714727610535245027, −4.96741094629270292043087887052, −2.89368209392102968926746241456, −0.792260187651210304738093363751, −0.01841941982553304146844166725,
2.23055881839537517848477698897, 3.59872900779479775344588534337, 4.76254768080969617228263661577, 6.09711091189663723408800746279, 7.68760699437085184511948587873, 8.013021814269027188921593879859, 9.914592918539349455048459536840, 10.23892816262778303727631460479, 11.10705644590249910975764246169, 11.93652311361917807851362850924