| L(s) = 1 | − 1.24·2-s + (−0.494 + 0.856i)3-s − 30.4·4-s + (24.3 − 42.1i)5-s + (0.616 − 1.06i)6-s + (83.1 + 144. i)7-s + 77.8·8-s + (121. + 209. i)9-s + (−30.3 + 52.6i)10-s − 436.·11-s + (15.0 − 26.0i)12-s + (486. + 842. i)13-s + (−103. − 179. i)14-s + (24.0 + 41.7i)15-s + 877.·16-s + (375. + 650. i)17-s + ⋯ |
| L(s) = 1 | − 0.220·2-s + (−0.0317 + 0.0549i)3-s − 0.951·4-s + (0.435 − 0.754i)5-s + (0.00699 − 0.0121i)6-s + (0.641 + 1.11i)7-s + 0.430·8-s + (0.497 + 0.862i)9-s + (−0.0960 + 0.166i)10-s − 1.08·11-s + (0.0301 − 0.0522i)12-s + (0.798 + 1.38i)13-s + (−0.141 − 0.244i)14-s + (0.0276 + 0.0478i)15-s + 0.856·16-s + (0.315 + 0.545i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.01800 + 0.647321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01800 + 0.647321i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.63e3 - 1.15e4i)T \) |
| good | 2 | \( 1 + 1.24T + 32T^{2} \) |
| 3 | \( 1 + (0.494 - 0.856i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-24.3 + 42.1i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-83.1 - 144. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + 436.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-486. - 842. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-375. - 650. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-283. + 490. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (498. - 863. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.99e3 + 3.46e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-3.89e3 + 6.74e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (6.02e3 - 1.04e4i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.55e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 8.22e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (2.10e3 - 3.64e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 - 3.81e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-1.40e4 - 2.44e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-5.91e3 + 1.02e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (3.77e4 + 6.53e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-2.78e4 - 4.81e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.89e4 + 6.74e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-8.25e3 + 1.42e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-3.81e4 + 6.61e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.06e3T + 8.58e9T^{2} \) |
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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
−15.21940670016295327547614478349, −13.67330804933204566164653181902, −13.10560430895601687360357447111, −11.61962641926300978387271134179, −10.05043793368702255477515560948, −8.886076081479075971360066429288, −8.025138311301018270954565804360, −5.56647025468892648988853561573, −4.58179402718384055270834952944, −1.73718995283540156506861307878,
0.800522247411870592612088235989, 3.56076196792263369072471550436, 5.30631301198236070707421642259, 7.17891100792717986070903455468, 8.408680100190340756780128416866, 10.16579632316849369501480135814, 10.58826703443503091558975895229, 12.63484614075823317561934900998, 13.67946705562467693775654654544, 14.49441191729158725608025306200
