L(s) = 1 | + (20.3 + 20.3i)3-s + (−46.4 + 31.1i)5-s + (76.9 − 76.9i)7-s + 588. i·9-s + 556. i·11-s + (141. − 141. i)13-s + (−1.58e3 − 312. i)15-s + (−477. − 477. i)17-s − 1.60e3·19-s + 3.13e3·21-s + (346. + 346. i)23-s + (1.19e3 − 2.88e3i)25-s + (−7.03e3 + 7.03e3i)27-s + 7.48e3i·29-s − 7.92e3i·31-s + ⋯ |
L(s) = 1 | + (1.30 + 1.30i)3-s + (−0.830 + 0.556i)5-s + (0.593 − 0.593i)7-s + 2.41i·9-s + 1.38i·11-s + (0.231 − 0.231i)13-s + (−1.81 − 0.359i)15-s + (−0.400 − 0.400i)17-s − 1.02·19-s + 1.55·21-s + (0.136 + 0.136i)23-s + (0.380 − 0.924i)25-s + (−1.85 + 1.85i)27-s + 1.65i·29-s − 1.48i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.04784 + 2.03553i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04784 + 2.03553i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (46.4 - 31.1i)T \) |
good | 3 | \( 1 + (-20.3 - 20.3i)T + 243iT^{2} \) |
| 7 | \( 1 + (-76.9 + 76.9i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 556. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-141. + 141. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (477. + 477. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 + 1.60e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-346. - 346. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 - 7.48e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.92e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-3.32e3 - 3.32e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.87e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-8.25e3 - 8.25e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-5.09e3 + 5.09e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.94e4 + 1.94e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 108.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.89e4 + 2.89e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 982. iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (2.15e4 - 2.15e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 9.38e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + (9.45e3 + 9.45e3i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 8.48e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.22e5 - 1.22e5i)T + 8.58e9iT^{2} \) |
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
−14.28566204090752030178819563145, −12.91634258368680686017742586586, −11.17381697650353837037112710656, −10.42973134224036858882389412601, −9.336722437049851539801693247715, −8.142916037404379424030031729037, −7.26618235554406992297635458881, −4.63902760207654684658125113893, −3.94121368514735196701443198747, −2.45270932408861420428421830277,
0.868767072660102397329778793329, 2.41472905427548108092111930753, 3.93205366615550259006113717626, 6.10512970579450500453458594354, 7.55668049530340336982832829289, 8.555762588687192347245163351584, 8.823430000939886900540510805789, 11.17782463503980968597477097836, 12.21465028574860771426039396794, 13.06432440905770726566286535471